Theorem fvelima 5693

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 5078	. . . 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 5678	. . . 4 |- ( Fun F -> ( x F A -> ( F ` x ) = A ) )
4	3	reximdv 2631	. . 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 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4086   "cima 4726   Fun wfun 5318   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  ssimaex  5703  ctssdccl  7301  suplocexprlemmu  7928  suplocexprlemloc  7931  ennnfonelemex  13025