Theorem fabexg 5405

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 4742	. 2 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
2		pwexg 4182	. 2 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
3		fabexg.1	. . . . 5 |- F = { x | ( x : A --> B /\ ph ) }
4		fssxp 5385	. . . . . . . 8 |- ( x : A --> B -> x C_ ( A X. B ) )
5		velpw 3584	. . . . . . . 8 |- ( x e. ~P ( A X. B ) <-> x C_ ( A X. B ) )
6	4, 5	sylibr 134	. . . . . . 7 |- ( x : A --> B -> x e. ~P ( A X. B ) )
7	6	anim1i 340	. . . . . 6 |- ( ( x : A --> B /\ ph ) -> ( x e. ~P ( A X. B ) /\ ph ) )
8	7	ss2abi 3229	. . . . 5 |- { x | ( x : A --> B /\ ph ) } C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
9	3, 8	eqsstri 3189	. . . 4 |- F C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
10		ssab2 3241	. . . 4 |- { x | ( x e. ~P ( A X. B ) /\ ph ) } C_ ~P ( A X. B )
11	9, 10	sstri 3166	. . 3 |- F C_ ~P ( A X. B )
12		ssexg 4144	. . 3 |- ( ( F C_ ~P ( A X. B ) /\ ~P ( A X. B ) e. _V ) -> F e. _V )
13	11, 12	mpan 424	. 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 104   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   X. cxp 4626   -->wf 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222

This theorem is referenced by:  fabex  5406  f1oabexg  5475