Theorem fabexg 7122

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 6960	. 2 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
2		pwexg 4850	. 2 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
3		fabexg.1	. . . . 5 |- F = { x | ( x : A --> B /\ ph ) }
4		fssxp 6060	. . . . . . . 8 |- ( x : A --> B -> x C_ ( A X. B ) )
5		selpw 4165	. . . . . . . 8 |- ( x e. ~P ( A X. B ) <-> x C_ ( A X. B ) )
6	4, 5	sylibr 224	. . . . . . 7 |- ( x : A --> B -> x e. ~P ( A X. B ) )
7	6	anim1i 592	. . . . . 6 |- ( ( x : A --> B /\ ph ) -> ( x e. ~P ( A X. B ) /\ ph ) )
8	7	ss2abi 3674	. . . . 5 |- { x | ( x : A --> B /\ ph ) } C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
9	3, 8	eqsstri 3635	. . . 4 |- F C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
10		ssab2 3686	. . . 4 |- { x | ( x e. ~P ( A X. B ) /\ ph ) } C_ ~P ( A X. B )
11	9, 10	sstri 3612	. . 3 |- F C_ ~P ( A X. B )
12		ssexg 4804	. . 3 |- ( ( F C_ ~P ( A X. B ) /\ ~P ( A X. B ) e. _V ) -> F e. _V )
13	11, 12	mpan 706	. 2 |- ( ~P ( A X. B ) e. _V -> F e. _V )
14	1, 2, 13	3syl 18	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fabex  7123  f1oabexg  7125  elghomlem1OLD  33684