Theorem mapex 7863

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 6060	. . . 4 |- ( f : A --> B -> f C_ ( A X. B ) )
2	1	ss2abi 3674	. . 3 |- { f | f : A --> B } C_ { f | f C_ ( A X. B ) }
3		df-pw 4160	. . 3 |- ~P ( A X. B ) = { f | f C_ ( A X. B ) }
4	2, 3	sseqtr4i 3638	. 2 |- { f | f : A --> B } C_ ~P ( A X. B )
5		xpexg 6960	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
6		pwexg 4850	. . 3 |- ( ( A X. B ) e. _V -> ~P ( A X. B ) e. _V )
7	5, 6	syl 17	. 2 |- ( ( A e. C /\ B e. D ) -> ~P ( A X. B ) e. _V )
8		ssexg 4804	. 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 695	1 |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fnmap  7864  mapvalg  7867  isghm  17660  wksfval  26505  measbase  30260  measval  30261  ismeas  30262  isrnmeas  30263  cnfex  39187  opabresexd  41306  upwlksfval  41716