Theorem fvelima 6248

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 5470	. . . 4 |- ( A e. ( F " B ) -> ( A e. ( F " B ) <-> E. x e. B x F A ) )
2	1	ibi 256	. . 3 |- ( A e. ( F " B ) -> E. x e. B x F A )
3		funbrfv 6234	. . . 4 |- ( Fun F -> ( x F A -> ( F ` x ) = A ) )
4	3	reximdv 3016	. . 3 |- ( Fun F -> ( E. x e. B x F A -> E. x e. B ( F ` x ) = A ) )
5	2, 4	syl5 34	. 2 |- ( Fun F -> ( A e. ( F " B ) -> E. x e. B ( F ` x ) = A ) )
6	5	imp 445	1 |- ( ( Fun F /\ A e. ( F " B ) ) -> E. x e. B ( F ` x ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   "cima 5117   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  ssimaex  6263  isofrlem  6590  tz7.49  7540  rankwflemb  8656  tcrank  8747  zorn2lem5  9322  zorn2lem6  9323  uniimadom  9366  wunr1om  9541  tskr1om  9589  tskr1om2  9590  grur1  9642  iscldtop  20899  kqfvima  21533  fmfnfmlem4  21761  fmfnfm  21762  qustgpopn  21923  c1liplem1  23759  plypf1  23968  ltgseg  25491  axcontlem9  25852  uhgrspan1  26195  pthdlem2lem  26663  htthlem  27774  xrofsup  29533  fimaproj  29900  txomap  29901  qtophaus  29903  erdszelem7  31179  erdszelem8  31180  mrsub0  31413  mrsubccat  31415  mrsubcn  31416  msubrn  31426  mthmblem  31477  ivthALT  32330  ftc2nc  33494  heibor1lem  33608  ismrc  37264  funimassd  39431  icccncfext  40100  dirkercncflem2  40321  smfpimbor1lem1  41005