Theorem rexrn 5653

Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Hypothesis

Ref	Expression
rexrn.1	|- ( x = ( F ` y ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexrn	|- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) )

Distinct variable groups:   x, y, A   x, F, y   ps, x   ph, y

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem rexrn

Step	Hyp	Ref	Expression
1		funfvex 5532	. . 3 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
2	1	funfni 5316	. 2 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
3		fvelrnb 5563	. . 3 |- ( F Fn A -> ( x e. ran F <-> E. y e. A ( F ` y ) = x ) )
4		eqcom 2179	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
5	4	rexbii 2484	. . 3 |- ( E. y e. A ( F ` y ) = x <-> E. y e. A x = ( F ` y ) )
6	3, 5	bitrdi 196	. 2 |- ( F Fn A -> ( x e. ran F <-> E. y e. A x = ( F ` y ) ) )
7		rexrn.1	. . 3 |- ( x = ( F ` y ) -> ( ph <-> ps ) )
8	7	adantl 277	. 2 |- ( ( F Fn A /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
9	2, 6, 8	rexxfr2d 4465	1 |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2737   ran crn 4627   Fn wfn 5211   ` cfv 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224

This theorem is referenced by:  elrnrexdm  5655  rexrnmpt  5659  cbvexfo  5786  rexanuz  10992  lmbr2  13607  lmff  13642