Theorem mapex 6644

Description: The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)

Assertion

Ref	Expression
mapex	|- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V )

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   C( f)   D( f)

Proof of Theorem mapex

Step	Hyp	Ref	Expression
1		fssxp 5375	. . . 4 |- ( f : A --> B -> f C_ ( A X. B ) )
2	1	ss2abi 3225	. . 3 |- { f | f : A --> B } C_ { f | f C_ ( A X. B ) }
3		df-pw 3574	. . 3 |- ~P ( A X. B ) = { f | f C_ ( A X. B ) }
4	2, 3	sseqtrri 3188	. 2 |- { f | f : A --> B } C_ ~P ( A X. B )
5		xpexg 4734	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
6		pwexg 4175	. . 3 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
7	5, 6	syl 14	. 2 |- ( ( A e. C /\ B e. D ) -> ~P ( A X. B ) e. _V )
8		ssexg 4137	. 2 |- ( ( { f | f : A --> B } C_ ~P ( A X. B ) /\ ~P ( A X. B ) e. _V ) -> { f | f : A --> B } e. _V )
9	4, 7, 8	sylancr 414	1 |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2146   {cab 2161   _Vcvv 2735   C_ wss 3127   ~Pcpw 3572   X. cxp 4618   -->wf 5204

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-f 5212

This theorem is referenced by:  fnmap  6645  mapvalg  6648  nninfex  7110  cnovex  13247  ispsmet  13374  cncfval  13610