Theorem fvelima 5546

Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
fvelima	|- ( ( Fun F /\ A e. ( F " B ) ) -> E. x e. B ( F ` x ) = A )

Distinct variable groups:   x, A   x, B   x, F

Proof of Theorem fvelima

Step	Hyp	Ref	Expression
1		elimag 4955	. . . 4 |- ( A e. ( F " B ) -> ( A e. ( F " B ) <-> E. x e. B x F A ) )
2	1	ibi 175	. . 3 |- ( A e. ( F " B ) -> E. x e. B x F A )
3		funbrfv 5533	. . . 4 |- ( Fun F -> ( x F A -> ( F ` x ) = A ) )
4	3	reximdv 2571	. . 3 |- ( Fun F -> ( E. x e. B x F A -> E. x e. B ( F ` x ) = A ) )
5	2, 4	syl5 32	. 2 |- ( Fun F -> ( A e. ( F " B ) -> E. x e. B ( F ` x ) = A ) )
6	5	imp 123	1 |- ( ( Fun F /\ A e. ( F " B ) ) -> E. x e. B ( F ` x ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3987   "cima 4612   Fun wfun 5190   ` cfv 5196

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fv 5204

This theorem is referenced by:  ssimaex  5555  ctssdccl  7084  suplocexprlemmu  7667  suplocexprlemloc  7670  ennnfonelemex  12356