:: Models and Satisfiability. Defining by Structural Induction and Free Variables in ZF-formulae
:: by Grzegorz Bancerek
::
:: Received April 14, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

scheme :: ZF_MODEL:sch 1
ZFschex{ F1( Variable, Variable) -> set , F2( Variable, Variable) -> set , F3( set ) -> set , F4( set , set ) -> set , F5( Variable, set ) -> set , F6() -> ZF-formula } :
ex a, A being set st
( ( for x, y being Variable holds
( [(x '=' y),F1(x,y)] in A & [(x 'in' y),F2(x,y)] in A ) ) & [F6(),a] in A & ( for H being ZF-formula
for a being set st [H,a] in A holds
( ( H is being_equality implies a = F1((Var1 H),(Var2 H)) ) & ( H is being_membership implies a = F2((Var1 H),(Var2 H)) ) & ( H is negative implies ex b being set st
( a = F3(b) & [(),b] in A ) ) & ( H is conjunctive implies ex b, c being set st
( a = F4(b,c) & [,b] in A & [,c] in A ) ) & ( H is universal implies ex b being set st
( a = F5((),b) & [(),b] in A ) ) ) ) )
proof end;

scheme :: ZF_MODEL:sch 2
ZFschuniq{ F1( Variable, Variable) -> set , F2( Variable, Variable) -> set , F3( set ) -> set , F4( set , set ) -> set , F5( Variable, set ) -> set , F6() -> ZF-formula, F7() -> set , F8() -> set } :
F7() = F8()
provided
A1: ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),F1(x,y)] in A & [(x 'in' y),F2(x,y)] in A ) ) & [F6(),F7()] in A & ( for H being ZF-formula
for a being set st [H,a] in A holds
( ( H is being_equality implies a = F1((Var1 H),(Var2 H)) ) & ( H is being_membership implies a = F2((Var1 H),(Var2 H)) ) & ( H is negative implies ex b being set st
( a = F3(b) & [(),b] in A ) ) & ( H is conjunctive implies ex b, c being set st
( a = F4(b,c) & [,b] in A & [,c] in A ) ) & ( H is universal implies ex b being set st
( a = F5((),b) & [(),b] in A ) ) ) ) ) and
A2: ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),F1(x,y)] in A & [(x 'in' y),F2(x,y)] in A ) ) & [F6(),F8()] in A & ( for H being ZF-formula
for a being set st [H,a] in A holds
( ( H is being_equality implies a = F1((Var1 H),(Var2 H)) ) & ( H is being_membership implies a = F2((Var1 H),(Var2 H)) ) & ( H is negative implies ex b being set st
( a = F3(b) & [(),b] in A ) ) & ( H is conjunctive implies ex b, c being set st
( a = F4(b,c) & [,b] in A & [,c] in A ) ) & ( H is universal implies ex b being set st
( a = F5((),b) & [(),b] in A ) ) ) ) )
proof end;

scheme :: ZF_MODEL:sch 3
ZFschresult{ F1( Variable, Variable) -> set , F2( Variable, Variable) -> set , F3( set ) -> set , F4( set , set ) -> set , F5( Variable, set ) -> set , F6() -> ZF-formula, F7( ZF-formula) -> set } :
( ( F6() is being_equality implies F7(F6()) = F1((Var1 F6()),(Var2 F6())) ) & ( F6() is being_membership implies F7(F6()) = F2((Var1 F6()),(Var2 F6())) ) & ( F6() is negative implies F7(F6()) = F3(F7((the_argument_of F6()))) ) & ( F6() is conjunctive implies for a, b being set st a = F7(()) & b = F7(()) holds
F7(F6()) = F4(a,b) ) & ( F6() is universal implies F7(F6()) = F5((bound_in F6()),F7((the_scope_of F6()))) ) )
provided
A1: for H9 being ZF-formula
for a being set holds
( a = F7(H9) iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),F1(x,y)] in A & [(x 'in' y),F2(x,y)] in A ) ) & [H9,a] in A & ( for H being ZF-formula
for a being set st [H,a] in A holds
( ( H is being_equality implies a = F1((Var1 H),(Var2 H)) ) & ( H is being_membership implies a = F2((Var1 H),(Var2 H)) ) & ( H is negative implies ex b being set st
( a = F3(b) & [(),b] in A ) ) & ( H is conjunctive implies ex b, c being set st
( a = F4(b,c) & [,b] in A & [,c] in A ) ) & ( H is universal implies ex b being set st
( a = F5((),b) & [(),b] in A ) ) ) ) ) )
proof end;

scheme :: ZF_MODEL:sch 4
ZFschproperty{ F1( Variable, Variable) -> set , F2( Variable, Variable) -> set , F3( set ) -> set , F4( set , set ) -> set , F5( Variable, set ) -> set , F6( ZF-formula) -> set , F7() -> ZF-formula, P1[ set ] } :
P1[F6(F7())]
provided
A1: for H9 being ZF-formula
for a being set holds
( a = F6(H9) iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),F1(x,y)] in A & [(x 'in' y),F2(x,y)] in A ) ) & [H9,a] in A & ( for H being ZF-formula
for a being set st [H,a] in A holds
( ( H is being_equality implies a = F1((Var1 H),(Var2 H)) ) & ( H is being_membership implies a = F2((Var1 H),(Var2 H)) ) & ( H is negative implies ex b being set st
( a = F3(b) & [(),b] in A ) ) & ( H is conjunctive implies ex b, c being set st
( a = F4(b,c) & [,b] in A & [,c] in A ) ) & ( H is universal implies ex b being set st
( a = F5((),b) & [(),b] in A ) ) ) ) ) ) and
A2: for x, y being Variable holds
( P1[F1(x,y)] & P1[F2(x,y)] ) and
A3: for a being set st P1[a] holds
P1[F3(a)] and
A4: for a, b being set st P1[a] & P1[b] holds
P1[F4(a,b)] and
A5: for a being set
for x being Variable st P1[a] holds
P1[F5(x,a)]
proof end;

deffunc H1( Variable, Variable) -> set = {$1,$2};

deffunc H2( Variable, Variable) -> set = {$1,$2};

deffunc H3( set ) -> set = $1; deffunc H4( set , set ) -> set = union {$1,$2}; deffunc H5( Variable, set ) -> Element of K27($2) = $2 \ {$1};

definition
let H be ZF-formula;
func Free H -> set means :Def1: :: ZF_MODEL:def 1
ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,it] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is being_membership implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is negative implies ex b being set st
( a = b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = union {b,c} & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = b \ {(bound_in H9)} & [(),b] in A ) ) ) ) );
existence
ex b1, A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b1] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is being_membership implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is negative implies ex b being set st
( a = b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = union {b,c} & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = b \ {(bound_in H9)} & [(),b] in A ) ) ) ) )
proof end;
uniqueness
for b1, b2 being set st ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b1] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is being_membership implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is negative implies ex b being set st
( a = b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = union {b,c} & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = b \ {(bound_in H9)} & [(),b] in A ) ) ) ) ) & ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b2] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is being_membership implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is negative implies ex b being set st
( a = b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = union {b,c} & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = b \ {(bound_in H9)} & [(),b] in A ) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Free ZF_MODEL:def 1 :
for H being ZF-formula
for b2 being set holds
( b2 = Free H iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b2] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is being_membership implies a = {(Var1 H9),(Var2 H9)} ) & ( H9 is negative implies ex b being set st
( a = b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = union {b,c} & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = b \ {(bound_in H9)} & [(),b] in A ) ) ) ) ) );

deffunc H6( ZF-formula) -> set = Free $1; Lm1: for H being ZF-formula for a1 being set holds ( a1 = H6(H) iff ex A being set st ( ( for x, y being Variable holds ( [(x '=' y),H1(x,y)] in A & [(x 'in' y),H2(x,y)] in A ) ) & [H,a1] in A & ( for H9 being ZF-formula for a being set st [H9,a] in A holds ( ( H9 is being_equality implies a = H1( Var1 H9, Var2 H9) ) & ( H9 is being_membership implies a = H2( Var1 H9, Var2 H9) ) & ( H9 is negative implies ex b being set st ( a = H3(b) & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st ( a = H4(b,c) & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st ( a = H5( bound_in H9,b) & [(),b] in A ) ) ) ) ) ) by Def1; Lm2: now :: thesis: for H being ZF-formula holds ( ( H is being_equality implies H6(H) = H1( Var1 H, Var2 H) ) & ( H is being_membership implies H6(H) = H2( Var1 H, Var2 H) ) & ( H is negative implies H6(H) = H3(H6( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H6( the_left_argument_of H) & b = H6( the_right_argument_of H) holds H6(H) = H4(a,b) ) & ( H is universal implies H6(H) = H5( bound_in H,H6( the_scope_of H)) ) ) let H be ZF-formula; :: thesis: ( ( H is being_equality implies H6(H) = H1( Var1 H, Var2 H) ) & ( H is being_membership implies H6(H) = H2( Var1 H, Var2 H) ) & ( H is negative implies H6(H) = H3(H6( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H6( the_left_argument_of H) & b = H6( the_right_argument_of H) holds H6(H) = H4(a,b) ) & ( H is universal implies H6(H) = H5( bound_in H,H6( the_scope_of H)) ) ) thus ( ( H is being_equality implies H6(H) = H1( Var1 H, Var2 H) ) & ( H is being_membership implies H6(H) = H2( Var1 H, Var2 H) ) & ( H is negative implies H6(H) = H3(H6( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H6( the_left_argument_of H) & b = H6( the_right_argument_of H) holds H6(H) = H4(a,b) ) & ( H is universal implies H6(H) = H5( bound_in H,H6( the_scope_of H)) ) ) from :: thesis: verum end; definition let H be ZF-formula; :: original: Free redefine func Free H -> Subset of VAR; coherence Free H is Subset of VAR proof end; end; theorem :: ZF_MODEL:1 for H being ZF-formula holds ( ( H is being_equality implies Free H = {(Var1 H),(Var2 H)} ) & ( H is being_membership implies Free H = {(Var1 H),(Var2 H)} ) & ( H is negative implies Free H = Free () ) & ( H is conjunctive implies Free H = () \/ () ) & ( H is universal implies Free H = (Free ()) \ {()} ) ) proof end; :: The set of all valuations of variables in a model definition let D be non empty set ; func VAL D -> set equals :: ZF_MODEL:def 2 Funcs (VAR,D); coherence Funcs (VAR,D) is set ; end; :: deftheorem defines VAL ZF_MODEL:def 2 : for D being non empty set holds VAL D = Funcs (VAR,D); registration let D be non empty set ; cluster VAL D -> non empty ; coherence not VAL D is empty ; end; :: The set of all valuations which satisfy a ZF-formula in a model definition deffunc H7( set , set ) -> set =$1 /\ $2; let H be ZF-formula; let E be non empty set ; deffunc H8( Variable, Variable) -> set = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds f .$1 = f . $2 } ; deffunc H9( Variable, Variable) -> set = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds f .$1 in f . $2 } ; deffunc H10( set ) -> Element of K27((VAL E)) = (VAL E) \$1;
deffunc H11( Variable, set ) -> set = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = $2 & f = v5 holds ( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds$1 = y ) holds
g in X ) )
}
;
func St (H,E) -> set means :Def3: :: ZF_MODEL:def 3
ex A being set st
( ( for x, y being Variable holds
( [(x '=' y), { v1 where v1 is Element of VAL E : for f being Function of VAR,E st f = v1 holds
f . x = f . y
}
]
in A & [(x 'in' y), { v2 where v2 is Element of VAL E : for f being Function of VAR,E st f = v2 holds
f . x in f . y
}
]
in A ) ) & [H,it] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . (Var1 H9) = f . (Var2 H9)
}
) & ( H9 is being_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR,E st f = v4 holds
f . (Var1 H9) in f . (Var2 H9)
}
) & ( H9 is negative implies ex b being set st
( a = (VAL E) \ b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = b /\ c & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H9 = y ) holds
g in X ) )
}
& [(),b] in A ) ) ) ) );
existence
ex b1, A being set st
( ( for x, y being Variable holds
( [(x '=' y), { v1 where v1 is Element of VAL E : for f being Function of VAR,E st f = v1 holds
f . x = f . y
}
]
in A & [(x 'in' y), { v2 where v2 is Element of VAL E : for f being Function of VAR,E st f = v2 holds
f . x in f . y
}
]
in A ) ) & [H,b1] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . (Var1 H9) = f . (Var2 H9)
}
) & ( H9 is being_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR,E st f = v4 holds
f . (Var1 H9) in f . (Var2 H9)
}
) & ( H9 is negative implies ex b being set st
( a = (VAL E) \ b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = b /\ c & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H9 = y ) holds
g in X ) )
}
& [(),b] in A ) ) ) ) )
proof end;
uniqueness
for b1, b2 being set st ex A being set st
( ( for x, y being Variable holds
( [(x '=' y), { v1 where v1 is Element of VAL E : for f being Function of VAR,E st f = v1 holds
f . x = f . y
}
]
in A & [(x 'in' y), { v2 where v2 is Element of VAL E : for f being Function of VAR,E st f = v2 holds
f . x in f . y
}
]
in A ) ) & [H,b1] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . (Var1 H9) = f . (Var2 H9)
}
) & ( H9 is being_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR,E st f = v4 holds
f . (Var1 H9) in f . (Var2 H9)
}
) & ( H9 is negative implies ex b being set st
( a = (VAL E) \ b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = b /\ c & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H9 = y ) holds
g in X ) )
}
& [(),b] in A ) ) ) ) ) & ex A being set st
( ( for x, y being Variable holds
( [(x '=' y), { v1 where v1 is Element of VAL E : for f being Function of VAR,E st f = v1 holds
f . x = f . y
}
]
in A & [(x 'in' y), { v2 where v2 is Element of VAL E : for f being Function of VAR,E st f = v2 holds
f . x in f . y
}
]
in A ) ) & [H,b2] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . (Var1 H9) = f . (Var2 H9)
}
) & ( H9 is being_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR,E st f = v4 holds
f . (Var1 H9) in f . (Var2 H9)
}
) & ( H9 is negative implies ex b being set st
( a = (VAL E) \ b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = b /\ c & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H9 = y ) holds
g in X ) )
}
& [(),b] in A ) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines St ZF_MODEL:def 3 :
for H being ZF-formula
for E being non empty set
for b3 being set holds
( b3 = St (H,E) iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y), { v1 where v1 is Element of VAL E : for f being Function of VAR,E st f = v1 holds
f . x = f . y
}
]
in A & [(x 'in' y), { v2 where v2 is Element of VAL E : for f being Function of VAR,E st f = v2 holds
f . x in f . y
}
]
in A ) ) & [H,b3] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . (Var1 H9) = f . (Var2 H9)
}
) & ( H9 is being_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR,E st f = v4 holds
f . (Var1 H9) in f . (Var2 H9)
}
) & ( H9 is negative implies ex b being set st
( a = (VAL E) \ b & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = b /\ c & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H9 = y ) holds
g in X ) )
}
& [(),b] in A ) ) ) ) ) );

Lm3: now :: thesis: for H being ZF-formula
for E being non empty set holds
( ( H is being_equality implies H12(H) = H8( Var1 H, Var2 H) ) & ( H is being_membership implies H12(H) = H9( Var1 H, Var2 H) ) & ( H is negative implies H12(H) = H10(H12( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H12( the_left_argument_of H) & b = H12( the_right_argument_of H) holds
H12(H) = H7(a,b) ) & ( H is universal implies H12(H) = H11( bound_in H,H12( the_scope_of H)) ) )
deffunc H7( set , set ) -> set = $1 /\$2;
let H be ZF-formula; :: thesis: for E being non empty set holds
( ( H is being_equality implies H12(H) = H8( Var1 H, Var2 H) ) & ( H is being_membership implies H12(H) = H9( Var1 H, Var2 H) ) & ( H is negative implies H12(H) = H10(H12( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H12( the_left_argument_of H) & b = H12( the_right_argument_of H) holds
H12(H) = H7(a,b) ) & ( H is universal implies H12(H) = H11( bound_in H,H12( the_scope_of H)) ) )

let E be non empty set ; :: thesis: ( ( H is being_equality implies H12(H) = H8( Var1 H, Var2 H) ) & ( H is being_membership implies H12(H) = H9( Var1 H, Var2 H) ) & ( H is negative implies H12(H) = H10(H12( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H12( the_left_argument_of H) & b = H12( the_right_argument_of H) holds
H12(H) = H7(a,b) ) & ( H is universal implies H12(H) = H11( bound_in H,H12( the_scope_of H)) ) )

deffunc H8( Variable, Variable) -> set = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . $1 = f .$2
}
;
deffunc H9( Variable, Variable) -> set = { v3 where v3 is Element of VAL E : for f being Function of VAR,E st f = v3 holds
f . $1 in f .$2
}
;
deffunc H10( set ) -> Element of K27((VAL E)) = (VAL E) \ $1; deffunc H11( Variable, set ) -> set = { v5 where v5 is Element of VAL E : for X being set for f being Function of VAR,E st X =$2 & f = v5 holds
( f in X & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
$1 = y ) holds g in X ) ) } ; deffunc H12( ZF-formula) -> set = St ($1,E);
A1: for H being ZF-formula
for a being set holds
( a = H12(H) iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),H8(x,y)] in A & [(x 'in' y),H9(x,y)] in A ) ) & [H,a] in A & ( for H9 being ZF-formula
for a being set st [H9,a] in A holds
( ( H9 is being_equality implies a = H8( Var1 H9, Var2 H9) ) & ( H9 is being_membership implies a = H9( Var1 H9, Var2 H9) ) & ( H9 is negative implies ex b being set st
( a = H10(b) & [(),b] in A ) ) & ( H9 is conjunctive implies ex b, c being set st
( a = H7(b,c) & [(),b] in A & [,c] in A ) ) & ( H9 is universal implies ex b being set st
( a = H11( bound_in H9,b) & [(),b] in A ) ) ) ) ) ) by Def3;
thus ( ( H is being_equality implies H12(H) = H8( Var1 H, Var2 H) ) & ( H is being_membership implies H12(H) = H9( Var1 H, Var2 H) ) & ( H is negative implies H12(H) = H10(H12( the_argument_of H)) ) & ( H is conjunctive implies for a, b being set st a = H12( the_left_argument_of H) & b = H12( the_right_argument_of H) holds
H12(H) = H7(a,b) ) & ( H is universal implies H12(H) = H11( bound_in H,H12( the_scope_of H)) ) ) from :: thesis: verum
end;

definition
let H be ZF-formula;
let E be non empty set ;
:: original: St
redefine func St (H,E) -> Subset of (VAL E);
coherence
St (H,E) is Subset of (VAL E)
proof end;
end;

theorem Th2: :: ZF_MODEL:2
for E being non empty set
for x, y being Variable
for f being Function of VAR,E holds
( f . x = f . y iff f in St ((x '=' y),E) )
proof end;

theorem Th3: :: ZF_MODEL:3
for E being non empty set
for x, y being Variable
for f being Function of VAR,E holds
( f . x in f . y iff f in St ((x 'in' y),E) )
proof end;

theorem Th4: :: ZF_MODEL:4
for E being non empty set
for H being ZF-formula
for f being Function of VAR,E holds
( not f in St (H,E) iff f in St ((),E) )
proof end;

theorem Th5: :: ZF_MODEL:5
for E being non empty set
for H, H9 being ZF-formula
for f being Function of VAR,E holds
( ( f in St (H,E) & f in St (H9,E) ) iff f in St ((H '&' H9),E) )
proof end;

theorem Th6: :: ZF_MODEL:6
for E being non empty set
for x being Variable
for H being ZF-formula
for f being Function of VAR,E holds
( ( f in St (H,E) & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
g in St (H,E) ) ) iff f in St ((All (x,H)),E) )
proof end;

theorem :: ZF_MODEL:7
for H being ZF-formula
for E being non empty set st H is being_equality holds
for f being Function of VAR,E holds
( f . (Var1 H) = f . (Var2 H) iff f in St (H,E) )
proof end;

theorem :: ZF_MODEL:8
for H being ZF-formula
for E being non empty set st H is being_membership holds
for f being Function of VAR,E holds
( f . (Var1 H) in f . (Var2 H) iff f in St (H,E) )
proof end;

theorem :: ZF_MODEL:9
for H being ZF-formula
for E being non empty set st H is negative holds
for f being Function of VAR,E holds
( not f in St ((),E) iff f in St (H,E) )
proof end;

theorem :: ZF_MODEL:10
for H being ZF-formula
for E being non empty set st H is conjunctive holds
for f being Function of VAR,E holds
( ( f in St (,E) & f in St (,E) ) iff f in St (H,E) )
proof end;

theorem :: ZF_MODEL:11
for H being ZF-formula
for E being non empty set st H is universal holds
for f being Function of VAR,E holds
( ( f in St ((),E) & ( for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
bound_in H = y ) holds
g in St ((),E) ) ) iff f in St (H,E) )
proof end;

:: The satisfaction of a ZF-formula in a model by a valuation
definition
let D be non empty set ;
let f be Function of VAR,D;
let H be ZF-formula;
pred D,f |= H means :Def4: :: ZF_MODEL:def 4
f in St (H,D);
end;

:: deftheorem Def4 defines |= ZF_MODEL:def 4 :
for D being non empty set
for f being Function of VAR,D
for H being ZF-formula holds
( D,f |= H iff f in St (H,D) );

theorem :: ZF_MODEL:12
for E being non empty set
for f being Function of VAR,E
for x, y being Variable holds
( E,f |= x '=' y iff f . x = f . y ) by Th2;

theorem :: ZF_MODEL:13
for E being non empty set
for f being Function of VAR,E
for x, y being Variable holds
( E,f |= x 'in' y iff f . x in f . y ) by Th3;

theorem Th14: :: ZF_MODEL:14
for E being non empty set
for f being Function of VAR,E
for H being ZF-formula holds
( E,f |= H iff not E,f |= 'not' H ) by Th4;

theorem Th15: :: ZF_MODEL:15
for E being non empty set
for f being Function of VAR,E
for H, H9 being ZF-formula holds
( E,f |= H '&' H9 iff ( E,f |= H & E,f |= H9 ) ) by Th5;

theorem Th16: :: ZF_MODEL:16
for E being non empty set
for f being Function of VAR,E
for H being ZF-formula
for x being Variable holds
( E,f |= All (x,H) iff for g being Function of VAR,E st ( for y being Variable st g . y <> f . y holds
x = y ) holds
E,g |= H )
proof end;

theorem :: ZF_MODEL:17
for E being non empty set
for f being Function of VAR,E
for H, H9 being ZF-formula holds
( E,f |= H 'or' H9 iff ( E,f |= H or E,f |= H9 ) )
proof end;

theorem Th18: :: ZF_MODEL:18
for E being non empty set
for f being Function of VAR,E
for H, H9 being ZF-formula holds
( E,f |= H => H9 iff ( E,f |= H implies E,f |= H9 ) )
proof end;

theorem :: ZF_MODEL:19
for E being non empty set
for f being Function of VAR,E
for H, H9 being ZF-formula holds
( E,f |= H <=> H9 iff ( E,f |= H iff E,f |= H9 ) )
proof end;

theorem Th20: :: ZF_MODEL:20
for E being non empty set
for f being Function of VAR,E
for H being ZF-formula
for x being Variable holds
( E,f |= Ex (x,H) iff ex g being Function of VAR,E st
( ( for y being Variable st g . y <> f . y holds
x = y ) & E,g |= H ) )
proof end;

theorem :: ZF_MODEL:21
for H being ZF-formula
for x, y being Variable
for E being non empty set
for f being Function of VAR,E holds
( E,f |= All (x,y,H) iff for g being Function of VAR,E st ( for z being Variable holds
( not g . z <> f . z or x = z or y = z ) ) holds
E,g |= H )
proof end;

theorem :: ZF_MODEL:22
for H being ZF-formula
for x, y being Variable
for E being non empty set
for f being Function of VAR,E holds
( E,f |= Ex (x,y,H) iff ex g being Function of VAR,E st
( ( for z being Variable holds
( not g . z <> f . z or x = z or y = z ) ) & E,g |= H ) )
proof end;

:: The satisfaction of a ZF-formula in a model
definition
let E be non empty set ;
let H be ZF-formula;
pred E |= H means :: ZF_MODEL:def 5
for f being Function of VAR,E holds E,f |= H;
end;

:: deftheorem defines |= ZF_MODEL:def 5 :
for E being non empty set
for H being ZF-formula holds
( E |= H iff for f being Function of VAR,E holds E,f |= H );

theorem :: ZF_MODEL:23
for H being ZF-formula
for x being Variable
for E being non empty set holds
( E |= All (x,H) iff E |= H )
proof end;

:: The axioms of ZF-language
definition
func the_axiom_of_extensionality -> ZF-formula equals :: ZF_MODEL:def 6
All ((),(x. 1),((All ((x. 2),(((x. 2) 'in' ()) <=> ((x. 2) 'in' (x. 1))))) => (() '=' (x. 1))));
correctness
coherence
All ((),(x. 1),((All ((x. 2),(((x. 2) 'in' ()) <=> ((x. 2) 'in' (x. 1))))) => (() '=' (x. 1)))) is ZF-formula
;
;
func the_axiom_of_pairs -> ZF-formula equals :: ZF_MODEL:def 7
All ((),(x. 1),(Ex ((x. 2),(All ((x. 3),(((x. 3) 'in' (x. 2)) <=> (((x. 3) '=' ()) 'or' ((x. 3) '=' (x. 1)))))))));
correctness
coherence
All ((),(x. 1),(Ex ((x. 2),(All ((x. 3),(((x. 3) 'in' (x. 2)) <=> (((x. 3) '=' ()) 'or' ((x. 3) '=' (x. 1))))))))) is ZF-formula
;
;
func the_axiom_of_unions -> ZF-formula equals :: ZF_MODEL:def 8
All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (Ex ((x. 3),(((x. 2) 'in' (x. 3)) '&' ((x. 3) 'in' ()))))))))));
correctness
coherence
All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (Ex ((x. 3),(((x. 2) 'in' (x. 3)) '&' ((x. 3) 'in' ())))))))))) is ZF-formula
;
;
func the_axiom_of_infinity -> ZF-formula equals :: ZF_MODEL:def 9
Ex ((),(x. 1),(((x. 1) 'in' ()) '&' (All ((x. 2),(((x. 2) 'in' ()) => (Ex ((x. 3),((((x. 3) 'in' ()) '&' ('not' ((x. 3) '=' (x. 2)))) '&' (All ((x. 4),(((x. 4) 'in' (x. 2)) => ((x. 4) 'in' (x. 3)))))))))))));
correctness
coherence
Ex ((),(x. 1),(((x. 1) 'in' ()) '&' (All ((x. 2),(((x. 2) 'in' ()) => (Ex ((x. 3),((((x. 3) 'in' ()) '&' ('not' ((x. 3) '=' (x. 2)))) '&' (All ((x. 4),(((x. 4) 'in' (x. 2)) => ((x. 4) 'in' (x. 3))))))))))))) is ZF-formula
;
;
func the_axiom_of_power_sets -> ZF-formula equals :: ZF_MODEL:def 10
All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (All ((x. 3),(((x. 3) 'in' (x. 2)) => ((x. 3) 'in' ()))))))))));
correctness
coherence
All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (All ((x. 3),(((x. 3) 'in' (x. 2)) => ((x. 3) 'in' ())))))))))) is ZF-formula
;
;
end;

:: deftheorem defines the_axiom_of_extensionality ZF_MODEL:def 6 :
the_axiom_of_extensionality = All ((),(x. 1),((All ((x. 2),(((x. 2) 'in' ()) <=> ((x. 2) 'in' (x. 1))))) => (() '=' (x. 1))));

:: deftheorem defines the_axiom_of_pairs ZF_MODEL:def 7 :
the_axiom_of_pairs = All ((),(x. 1),(Ex ((x. 2),(All ((x. 3),(((x. 3) 'in' (x. 2)) <=> (((x. 3) '=' ()) 'or' ((x. 3) '=' (x. 1)))))))));

:: deftheorem defines the_axiom_of_unions ZF_MODEL:def 8 :
the_axiom_of_unions = All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (Ex ((x. 3),(((x. 2) 'in' (x. 3)) '&' ((x. 3) 'in' ()))))))))));

:: deftheorem defines the_axiom_of_infinity ZF_MODEL:def 9 :
the_axiom_of_infinity = Ex ((),(x. 1),(((x. 1) 'in' ()) '&' (All ((x. 2),(((x. 2) 'in' ()) => (Ex ((x. 3),((((x. 3) 'in' ()) '&' ('not' ((x. 3) '=' (x. 2)))) '&' (All ((x. 4),(((x. 4) 'in' (x. 2)) => ((x. 4) 'in' (x. 3)))))))))))));

:: deftheorem defines the_axiom_of_power_sets ZF_MODEL:def 10 :
the_axiom_of_power_sets = All ((),(Ex ((x. 1),(All ((x. 2),(((x. 2) 'in' (x. 1)) <=> (All ((x. 3),(((x. 3) 'in' (x. 2)) => ((x. 3) 'in' ()))))))))));

definition
let H be ZF-formula;
func the_axiom_of_substitution_for H -> ZF-formula equals :: ZF_MODEL:def 11
(All ((x. 3),(Ex ((),(All ((x. 4),(H <=> ((x. 4) '=' ())))))))) => (All ((x. 1),(Ex ((x. 2),(All ((x. 4),(((x. 4) 'in' (x. 2)) <=> (Ex ((x. 3),(((x. 3) 'in' (x. 1)) '&' H))))))))));
correctness
coherence
(All ((x. 3),(Ex ((),(All ((x. 4),(H <=> ((x. 4) '=' ())))))))) => (All ((x. 1),(Ex ((x. 2),(All ((x. 4),(((x. 4) 'in' (x. 2)) <=> (Ex ((x. 3),(((x. 3) 'in' (x. 1)) '&' H)))))))))) is ZF-formula
;
;
end;

:: deftheorem defines the_axiom_of_substitution_for ZF_MODEL:def 11 :
for H being ZF-formula holds the_axiom_of_substitution_for H = (All ((x. 3),(Ex ((),(All ((x. 4),(H <=> ((x. 4) '=' ())))))))) => (All ((x. 1),(Ex ((x. 2),(All ((x. 4),(((x. 4) 'in' (x. 2)) <=> (Ex ((x. 3),(((x. 3) 'in' (x. 1)) '&' H))))))))));

definition
let E be non empty set ;
attr E is being_a_model_of_ZF means :: ZF_MODEL:def 12
( E is epsilon-transitive & E |= the_axiom_of_pairs & E |= the_axiom_of_unions & E |= the_axiom_of_infinity & E |= the_axiom_of_power_sets & ( for H being ZF-formula st {(),(x. 1),(x. 2)} misses Free H holds
E |= the_axiom_of_substitution_for H ) );
end;

:: deftheorem defines being_a_model_of_ZF ZF_MODEL:def 12 :
for E being non empty set holds
( E is being_a_model_of_ZF iff ( E is epsilon-transitive & E |= the_axiom_of_pairs & E |= the_axiom_of_unions & E |= the_axiom_of_infinity & E |= the_axiom_of_power_sets & ( for H being ZF-formula st {(),(x. 1),(x. 2)} misses Free H holds
E |= the_axiom_of_substitution_for H ) ) );