Theorem fvelima 5568

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 4975	. . . 4 |- ( A e. ( F " B ) -> ( A e. ( F " B ) <-> E. x e. B x F A ) )
2	1	ibi 176	. . 3 |- ( A e. ( F " B ) -> E. x e. B x F A )
3		funbrfv 5555	. . . 4 |- ( Fun F -> ( x F A -> ( F ` x ) = A ) )
4	3	reximdv 2578	. . 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 124	1 |- ( ( Fun F /\ A e. ( F " B ) ) -> E. x e. B ( F ` x ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4004   "cima 4630   Fun wfun 5211   ` cfv 5217

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 4122  ax-pow 4175  ax-pr 4210

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 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fv 5225

This theorem is referenced by:  ssimaex  5578  ctssdccl  7110  suplocexprlemmu  7717  suplocexprlemloc  7720  ennnfonelemex  12415