let Omega, Omega2 be non empty set ; :: thesis: for F being Function of Omega,Omega2

for y being non empty set holds { z where z is Element of Omega : F . z is Element of y } = F " y

let F be Function of Omega,Omega2; :: thesis: for y being non empty set holds { z where z is Element of Omega : F . z is Element of y } = F " y

let y be non empty set ; :: thesis: { z where z is Element of Omega : F . z is Element of y } = F " y

set D = { z where z is Element of Omega : F . z is Element of y } ;

for x being object holds

( x in { z where z is Element of Omega : F . z is Element of y } iff x in F " y )

for y being non empty set holds { z where z is Element of Omega : F . z is Element of y } = F " y

let F be Function of Omega,Omega2; :: thesis: for y being non empty set holds { z where z is Element of Omega : F . z is Element of y } = F " y

let y be non empty set ; :: thesis: { z where z is Element of Omega : F . z is Element of y } = F " y

set D = { z where z is Element of Omega : F . z is Element of y } ;

for x being object holds

( x in { z where z is Element of Omega : F . z is Element of y } iff x in F " y )

proof

hence
{ z where z is Element of Omega : F . z is Element of y } = F " y
by TARSKI:2; :: thesis: verum
let x be object ; :: thesis: ( x in { z where z is Element of Omega : F . z is Element of y } iff x in F " y )

then A3: ( x in dom F & F . x in y ) by FUNCT_1:def 7;

thus x in { z where z is Element of Omega : F . z is Element of y } by A3; :: thesis: verum

end;hereby :: thesis: ( x in F " y implies x in { z where z is Element of Omega : F . z is Element of y } )

assume
x in F " y
; :: thesis: x in { z where z is Element of Omega : F . z is Element of y } assume
x in { z where z is Element of Omega : F . z is Element of y }
; :: thesis: x in F " y

then consider z being Element of Omega such that

A1: ( x = z & F . z is Element of y ) ;

z in Omega ;

then A2: z in dom F by FUNCT_2:def 1;

F . z in y by A1;

then z in F " y by FUNCT_1:def 7, A2;

hence x in F " y by A1; :: thesis: verum

end;then consider z being Element of Omega such that

A1: ( x = z & F . z is Element of y ) ;

z in Omega ;

then A2: z in dom F by FUNCT_2:def 1;

F . z in y by A1;

then z in F " y by FUNCT_1:def 7, A2;

hence x in F " y by A1; :: thesis: verum

then A3: ( x in dom F & F . x in y ) by FUNCT_1:def 7;

thus x in { z where z is Element of Omega : F . z is Element of y } by A3; :: thesis: verum