Theorem mapex 6636

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 5367	. . . 4 |- ( f : A --> B -> f C_ ( A X. B ) )
2	1	ss2abi 3220	. . 3 |- { f | f : A --> B } C_ { f | f C_ ( A X. B ) }
3		df-pw 3569	. . 3 |- ~P ( A X. B ) = { f | f C_ ( A X. B ) }
4	2, 3	sseqtrri 3183	. 2 |- { f | f : A --> B } C_ ~P ( A X. B )
5		xpexg 4726	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
6		pwexg 4167	. . 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 4129	. 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 412	1 |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2142   {cab 2157   _Vcvv 2731   C_ wss 3122   ~Pcpw 3567   X. cxp 4610   -->wf 5196

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-xp 4618  df-rel 4619  df-cnv 4620  df-dm 4622  df-rn 4623  df-fun 5202  df-fn 5203  df-f 5204

This theorem is referenced by:  fnmap  6637  mapvalg  6640  nninfex  7102  cnovex  13075  ispsmet  13202  cncfval  13438