Theorem fvelima 4723

Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (The proof was 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		simpr 350	. 2 |- ((Fun F /\ A e. (F"B)) -> A e. (F"B))
2		funopfvg 4711	. . . . . 6 |- ((A e. (F"B) /\ Fun F) -> (<.x, A>. e. F -> (F` x) = A))
3		df-br 3339	. . . . . 6 |- (xFA <-> <.x, A>. e. F)
4	2, 3	syl5ib 223	. . . . 5 |- ((A e. (F"B) /\ Fun F) -> (xFA -> (F` x) = A))
5	4	ancoms 484	. . . 4 |- ((Fun F /\ A e. (F"B)) -> (xFA -> (F` x) = A))
6	5	reximdv 2202	. . 3 |- ((Fun F /\ A e. (F"B)) -> (E.x e. B xFA -> E.x e. B (F` x) = A))
7		elimag 4269	. . . 4 |- (A e. (F"B) -> (A e. (F"B) <-> E.x e. B xFA))
8	7	ibi 652	. . 3 |- (A e. (F"B) -> E.x e. B xFA)
9	6, 8	syl5 20	. 2 |- ((Fun F /\ A e. (F"B)) -> (A e. (F"B) -> E.x e. B (F` x) = A))
10	1, 9	mpd 29	1 |- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  <.cop 3046   class class class wbr 3338  "cima 3989  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  ssimaex 4729  isofrlem 4878  tz7.49 5168  ordtypelem6 5689  ordtypelem7 5690  zorn2lem5 5954  zorn2lem6 5955  uniimadom 5972  ordtypelem6OLD 15380  ordtypelem7OLD 15381  ivthALT 15454  fmfnfmlem4 15597  fmfnfm 15598  filnetlem5 15644  filnet 15645  ismtyhmeolem 15950  heiborlem10 15964  heiborlem13 15967  heiborlem15 15969

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014

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