Theorem fabexg 5375

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 4718	. 2 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
2		pwexg 4159	. 2 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
3		fabexg.1	. . . . 5 |- F = { x | ( x : A --> B /\ ph ) }
4		fssxp 5355	. . . . . . . 8 |- ( x : A --> B -> x C_ ( A X. B ) )
5		velpw 3566	. . . . . . . 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 3214	. . . . 5 |- { x | ( x : A --> B /\ ph ) } C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
9	3, 8	eqsstri 3174	. . . 4 |- F C_ { x | ( x e. ~P ( A X. B ) /\ ph ) }
10		ssab2 3226	. . . 4 |- { x | ( x e. ~P ( A X. B ) /\ ph ) } C_ ~P ( A X. B )
11	9, 10	sstri 3151	. . 3 |- F C_ ~P ( A X. B )
12		ssexg 4121	. . 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 1343   e. wcel 2136   {cab 2151   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   X. cxp 4602   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  fabex  5376  f1oabexg  5444