Theorem fvelima 2859

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 group(s):   x,A   x,B   x,F

Proof of Theorem fvelima

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 |- (y = A -> (y e. (F"B) <-> A e. (F"B)))
2		cleq2 1110	. . . . . . 7 |- (y = A -> ((F` x) = y <-> (F` x) = A))
3	2	birexdv 1220	. . . . . 6 |- (y = A -> (E.x e. B (F` x) = y <-> E.x e. B (F` x) = A))
4	1, 3	imbi12d 474	. . . . 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 464	. . . 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 1350	. . . . . . . 8 |- y e. V
7	6	funfvopi 2853	. . . . . . 7 |- (Fun F -> (<.x, y>. e. F -> (F` x) = y))
8		df-br 2063	. . . . . . 7 |- (xFy <-> <.x, y>. e. F)
9	7, 8	syl5ib 181	. . . . . 6 |- (Fun F -> (xFy -> (F` x) = y))
10	9	r19.22sdv 1279	. . . . 5 |- (Fun F -> (E.x e. B xFy -> E.x e. B (F` x) = y))
11	6	elima 2606	. . . . 5 |- (y e. (F"B) <-> E.x e. B xFy)
12	10, 11	syl5ib 181	. . . 4 |- (Fun F -> (y e. (F"B) -> E.x e. B (F` x) = y))
13	5, 12	vtoclg 1383	. . 3 |- (A e. (F"B) -> (Fun F -> (A e. (F"B) -> E.x e. B (F` x) = A)))
14	13	pm2.43b 61	. 2 |- (Fun F -> (A e. (F"B) -> E.x e. B (F` x) = A))
15	14	imp 277	1 |- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202  <.cop 1810   class class class wbr 2054  "cima 2413  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  isofrlem 2939  tz7.49 2997  zornlem5 3607  zornlem6 3608

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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