Theorem fabexg 5485

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 4807	. 2 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
2		pwexg 4240	. 2 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
3		fabexg.1	. . . . 5 |- F = { x | ( x : A --> B /\ ph ) }
4		fssxp 5463	. . . . . . . 8 |- ( x : A --> B -> x C_ ( A X. B ) )
5		velpw 3633	. . . . . . . 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 3273	. . . . 5 |- { x | ( x : A --> B /\ ph ) } C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
9	3, 8	eqsstri 3233	. . . 4 |- F C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
10		ssab2 3285	. . . 4 |- { x | ( x e. ~P ( A X. B ) /\ ph ) } C_ ~P ( A X. B )
11	9, 10	sstri 3210	. . 3 |- F C_ ~P ( A X. B )
12		ssexg 4199	. . 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 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   X. cxp 4691   -->wf 5286

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-fun 5292  df-fn 5293  df-f 5294

This theorem is referenced by:  fabex  5486  f1oabexg  5556