Theorem fvelima 5481

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 4893	. . . 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 5468	. . . 4 |- ( Fun F -> ( x F A -> ( F ` x ) = A ) )
4	3	reximdv 2536	. . 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 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937   "cima 4550   Fun wfun 5125   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by:  ssimaex  5490  ctssdccl  7004  suplocexprlemmu  7550  suplocexprlemloc  7553  ennnfonelemex  11963