Theorem mapex 6731

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 5437	. . . 4 |- ( f : A --> B -> f C_ ( A X. B ) )
2	1	ss2abi 3264	. . 3 |- { f | f : A --> B } C_ { f | f C_ ( A X. B ) }
3		df-pw 3617	. . 3 |- ~P ( A X. B ) = { f | f C_ ( A X. B ) }
4	2, 3	sseqtrri 3227	. 2 |- { f | f : A --> B } C_ ~P ( A X. B )
5		xpexg 4787	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
6		pwexg 4223	. . 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 4182	. 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 2175   {cab 2190   _Vcvv 2771   C_ wss 3165   ~Pcpw 3615   X. cxp 4671   -->wf 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4679  df-rel 4680  df-cnv 4681  df-dm 4683  df-rn 4684  df-fun 5270  df-fn 5271  df-f 5272

This theorem is referenced by:  fnmap  6732  mapvalg  6735  exmidpw2en  6991  nninfex  7205  ptex  13014  isghm  13497  psrval  14346  psrbasg  14354  cnovex  14586  ispsmet  14713  cncfval  14962