Theorem fabexg 4596

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 4095	. 2 |- ((A e. C /\ B e. D) -> (A X. B) e. _V)
2		pwexg 3489	. 2 |- ((A X. B) e. _V -> ~P(A X. B) e. _V)
3		fabexg.1	. . . . 5 |- F = {x | (x:A-->B /\ ph)}
4		fssxp 4575	. . . . . . . 8 |- (x:A-->B -> x C_ (A X. B))
5		visset 2295	. . . . . . . . 9 |- x e. _V
6	5	elpw 3037	. . . . . . . 8 |- (x e. ~P(A X. B) <-> x C_ (A X. B))
7	4, 6	sylibr 217	. . . . . . 7 |- (x:A-->B -> x e. ~P(A X. B))
8	7	anim1i 361	. . . . . 6 |- ((x:A-->B /\ ph) -> (x e. ~P(A X. B) /\ ph))
9	8	ss2abi 2679	. . . . 5 |- {x | (x:A-->B /\ ph)} C_ {x | (x e. ~P(A X. B) /\ ph)}
10	3, 9	eqsstri 2647	. . . 4 |- F C_ {x | (x e. ~P(A X. B) /\ ph)}
11		ssab2 2691	. . . 4 |- {x | (x e. ~P(A X. B) /\ ph)} C_ ~P(A X. B)
12	10, 11	sstri 2626	. . 3 |- F C_ ~P(A X. B)
13		ssexg 3457	. . 3 |- ((F C_ ~P(A X. B) /\ ~P(A X. B) e. _V) -> F e. _V)
14	12, 13	mpan 759	. 2 |- (~P(A X. B) e. _V -> F e. _V)
15	1, 2, 14	3syl 24	1 |- ((A e. C /\ B e. D) -> F e. _V)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032   X. cxp 3984  -->wf 3994

This theorem is referenced by:  fabex 4597  f1oabexg 4650  elghomlem1 10193

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-13 1311  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  ax-un 3790

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-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-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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