Theorem fabexg 3653

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
fabexg.1	|- F = {x | (x:A-->B /\ ph)}

Assertion

Ref	Expression
fabexg	|- ((A e. C /\ B e. D) -> F e. V)

Distinct variable groups:   x,A   x,B

Proof of Theorem fabexg

Step	Hyp	Ref	Expression
1		xpexg 3259	. 2 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
2		pwexg 2746	. 2 |- ((A X. B) e. V -> P~(A X. B) e. V)
3		fabexg.1	. . . . 5 |- F = {x | (x:A-->B /\ ph)}
4		fssxp 3637	. . . . . . . 8 |- (x:A-->B -> x (_ (A X. B))
5		visset 1813	. . . . . . . . 9 |- x e. V
6	5	elpw 2404	. . . . . . . 8 |- (x e. P~(A X. B) <-> x (_ (A X. B))
7	4, 6	sylibr 200	. . . . . . 7 |- (x:A-->B -> x e. P~(A X. B))
8	7	anim1i 334	. . . . . 6 |- ((x:A-->B /\ ph) -> (x e. P~(A X. B) /\ ph))
9	8	ss2abi 2120	. . . . 5 |- {x | (x:A-->B /\ ph)} (_ {x | (x e. P~(A X. B) /\ ph)}
10	3, 9	eqsstr 2091	. . . 4 |- F (_ {x | (x e. P~(A X. B) /\ ph)}
11		ssab2 2130	. . . 4 |- {x | (x e. P~(A X. B) /\ ph)} (_ P~(A X. B)
12	10, 11	sstri 2073	. . 3 |- F (_ P~(A X. B)
13		ssexg 2721	. . 3 |- ((F (_ P~(A X. B) /\ P~(A X. B) e. V) -> F e. V)
14	12, 13	mpan 695	. 2 |- (P~(A X. B) e. V -> F e. V)
15	1, 2, 14	3syl 20	1 |- ((A e. C /\ B e. D) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047  P~cpw 2401   X. cxp 3168  -->wf 3178

This theorem is referenced by:  fabex 3654  f1oabexg 3700  elghomlem1 10382

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-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  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-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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