Theorem rexrn 5550

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 5431	. . 3 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
2	1	funfni 5218	. 2 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
3		fvelrnb 5462	. . 3 |- ( F Fn A -> ( x e. ran F <-> E. y e. A ( F ` y ) = x ) )
4		eqcom 2139	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
5	4	rexbii 2440	. . 3 |- ( E. y e. A ( F ` y ) = x <-> E. y e. A x = ( F ` y ) )
6	3, 5	syl6bb 195	. 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 275	. 2 |- ( ( F Fn A /\ x = ( F ` y ) ) -> ( ph <-> ps ) )
9	2, 6, 8	rexxfr2d 4381	1 |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2415   _Vcvv 2681   ran crn 4535   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  elrnrexdm  5552  rexrnmpt  5556  cbvexfo  5680  rexanuz  10753  lmbr2  12372  lmff  12407