Theorem ssimaexg 3769

Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Assertion

Ref	Expression
ssimaexg	|- ((A e. C /\ Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x)))

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem ssimaexg

Step	Hyp	Ref	Expression
1		imaeq2 3402	. . . . . 6 |- (y = A -> (F"y) = (F"A))
2	1	sseq2d 2089	. . . . 5 |- (y = A -> (B (_ (F"y) <-> B (_ (F"A)))
3	2	anbi2d 616	. . . 4 |- (y = A -> ((Fun F /\ B (_ (F"y)) <-> (Fun F /\ B (_ (F"A))))
4		sseq2 2083	. . . . . 6 |- (y = A -> (x (_ y <-> x (_ A))
5	4	anbi1d 617	. . . . 5 |- (y = A -> ((x (_ y /\ B = (F"x)) <-> (x (_ A /\ B = (F"x))))
6	5	exbidv 1279	. . . 4 |- (y = A -> (E.x(x (_ y /\ B = (F"x)) <-> E.x(x (_ A /\ B = (F"x))))
7	3, 6	imbi12d 626	. . 3 |- (y = A -> (((Fun F /\ B (_ (F"y)) -> E.x(x (_ y /\ B = (F"x))) <-> ((Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x)))))
8		visset 1813	. . . 4 |- y e. V
9	8	ssimaex 3768	. . 3 |- ((Fun F /\ B (_ (F"y)) -> E.x(x (_ y /\ B = (F"x)))
10	7, 9	vtoclg 1847	. 2 |- (A e. C -> ((Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x))))
11	10	3impib 831	1 |- ((A e. C /\ Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980   (_ wss 2047  "cima 3173  Fun wfun 3176

This theorem is referenced by:  subtop 7646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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