let R, S be Relation; :: thesis: for F being Function st F is_isomorphism_of R,S holds
F " is_isomorphism_of S,R
let F be Function; :: thesis: ( F is_isomorphism_of R,S implies F " is_isomorphism_of S,R )
assume A1: F is_isomorphism_of R,S ; :: thesis: F " is_isomorphism_of S,R
then A2: F is one-to-one by Def7;
A3: rng F = field S by A1, Def7;
hence A4: dom (F " ) = field S by A2, FUNCT_1:55; :: according to WELLORD1:def 7 :: thesis: ( rng (F " ) = field R & F " is one-to-one & ( for a, b being set holds
( [a,b] in S iff ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R ) ) ) )
A5: dom F = field R by A1, Def7;
hence rng (F " ) = field R by A2, FUNCT_1:55; :: thesis: ( F " is one-to-one & ( for a, b being set holds
( [a,b] in S iff ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R ) ) ) )
thus F " is one-to-one by A2; :: thesis: for a, b being set holds
( [a,b] in S iff ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R ) )
let a, b be set ; :: thesis: ( [a,b] in S iff ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R ) )
thus ( [a,b] in S implies ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R ) ) :: thesis: ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R implies [a,b] in S )

proof

A6: dom F = rng (F " ) by A2, FUNCT_1:55;
assume A7: [a,b] in S ; :: thesis: ( a in field S & b in field S & [((F " ) . a),((F " ) . b)] in R )
hence A8: ( a in field S & b in field S ) by RELAT_1:30; :: thesis: [((F " ) . a),((F " ) . b)] in R
then A9: ( (F " ) . a in rng (F " ) & (F " ) . b in rng (F " ) ) by A4, FUNCT_1:def 5;
( a = F . ((F " ) . a) & b = F . ((F " ) . b) ) by A3, A2, A8, FUNCT_1:57;
hence [((F " ) . a),((F " ) . b)] in R by A1, A5, A7, A6, A9, Def7; :: thesis: verum

end;

assume that
A10: ( a in field S & b in field S ) and
A11: [((F " ) . a),((F " ) . b)] in R ; :: thesis: [a,b] in S
( F . ((F " ) . a) = a & F . ((F " ) . b) = b ) by A3, A2, A10, FUNCT_1:57;
hence [a,b] in S by A1, A11, Def7; :: thesis: verum