Theorem fabexg 5356

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

Allowed substitution hints:   ph( x)   C( x)   D( x)   F( x)

Proof of Theorem fabexg

Step	Hyp	Ref	Expression
1		xpexg 4699	. 2 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
2		pwexg 4141	. 2 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
3		fabexg.1	. . . . 5 |- F = { x | ( x : A --> B /\ ph ) }
4		fssxp 5336	. . . . . . . 8 |- ( x : A --> B -> x C_ ( A X. B ) )
5		velpw 3550	. . . . . . . 8 |- ( x e. ~P ( A X. B ) <-> x C_ ( A X. B ) )
6	4, 5	sylibr 133	. . . . . . 7 |- ( x : A --> B -> x e. ~P ( A X. B ) )
7	6	anim1i 338	. . . . . 6 |- ( ( x : A --> B /\ ph ) -> ( x e. ~P ( A X. B ) /\ ph ) )
8	7	ss2abi 3200	. . . . 5 |- { x | ( x : A --> B /\ ph ) } C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
9	3, 8	eqsstri 3160	. . . 4 |- F C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
10		ssab2 3212	. . . 4 |- { x | ( x e. ~P ( A X. B ) /\ ph ) } C_ ~P ( A X. B )
11	9, 10	sstri 3137	. . 3 |- F C_ ~P ( A X. B )
12		ssexg 4103	. . 3 |- ( ( F C_ ~P ( A X. B ) /\ ~P ( A X. B ) e. _V ) -> F e. _V )
13	11, 12	mpan 421	. 2 |- ( ~P ( A X. B ) e. _V -> F e. _V )
14	1, 2, 13	3syl 17	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   {cab 2143   _Vcvv 2712   C_ wss 3102   ~Pcpw 3543   X. cxp 4583   -->wf 5165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596  df-fun 5171  df-fn 5172  df-f 5173

This theorem is referenced by:  fabex  5357  f1oabexg  5425