Theorem rexrn 5622

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 5503	. . 3 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. _V )
2	1	funfni 5288	. 2 |- ( ( F Fn A /\ y e. A ) -> ( F ` y ) e. _V )
3		fvelrnb 5534	. . 3 |- ( F Fn A -> ( x e. ran F <-> E. y e. A ( F ` y ) = x ) )
4		eqcom 2167	. . . 4 |- ( ( F ` y ) = x <-> x = ( F ` y ) )
5	4	rexbii 2473	. . 3 |- ( E. y e. A ( F ` y ) = x <-> E. y e. A x = ( F ` y ) )
6	3, 5	bitrdi 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 4443	1 |- ( F Fn A -> ( E. x e. ran F ph <-> E. y e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   _Vcvv 2726   ran crn 4605   Fn wfn 5183   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  elrnrexdm  5624  rexrnmpt  5628  cbvexfo  5754  rexanuz  10930  lmbr2  12854  lmff  12889