Theorem fvelima 3761

Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.

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		eleq1 1533	. . . . . 6 |- (y = A -> (y e. (F"B) <-> A e. (F"B)))
2		eqeq2 1483	. . . . . . 7 |- (y = A -> ((F` x) = y <-> (F` x) = A))
3	2	rexbidv 1663	. . . . . 6 |- (y = A -> (E.x e. B (F` x) = y <-> E.x e. B (F` x) = A))
4	1, 3	imbi12d 625	. . . . 5 |- (y = A -> ((y e. (F"B) -> E.x e. B (F` x) = y) <-> (A e. (F"B) -> E.x e. B (F` x) = A)))
5	4	imbi2d 611	. . . 4 |- (y = A -> ((Fun F -> (y e. (F"B) -> E.x e. B (F` x) = y)) <-> (Fun F -> (A e. (F"B) -> E.x e. B (F` x) = A))))
6		visset 1811	. . . . . . 7 |- y e. V
7	6	funbrfv 3747	. . . . . 6 |- (Fun F -> (xFy -> (F` x) = y))
8	7	r19.22sdv 1737	. . . . 5 |- (Fun F -> (E.x e. B xFy -> E.x e. B (F` x) = y))
9	6	elima 3405	. . . . 5 |- (y e. (F"B) <-> E.x e. B xFy)
10	8, 9	syl5ib 206	. . . 4 |- (Fun F -> (y e. (F"B) -> E.x e. B (F` x) = y))
11	5, 10	vtoclg 1845	. . 3 |- (A e. (F"B) -> (Fun F -> (A e. (F"B) -> E.x e. B (F` x) = A)))
12	11	pm2.43b 67	. 2 |- (Fun F -> (A e. (F"B) -> E.x e. B (F` x) = A))
13	12	imp 350	1 |- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E.wrex 1645   class class class wbr 2616  "cima 3170  Fun wfun 3173  ` cfv 3179

This theorem is referenced by:  ssimaex 3765  isofrlem 3898  tz7.49 3956  zorn2lem5 4779  zorn2lem6 4780  uniimadom 4797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fv 3195

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