Theorem mapex 6866

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 5510	. . . 4 |- ( f : A --> B -> f C_ ( A X. B ) )
2	1	ss2abi 3300	. . 3 |- { f | f : A --> B } C_ { f | f C_ ( A X. B ) }
3		df-pw 3658	. . 3 |- ~P ( A X. B ) = { f | f C_ ( A X. B ) }
4	2, 3	sseqtrri 3263	. 2 |- { f | f : A --> B } C_ ~P ( A X. B )
5		xpexg 4846	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
6		pwexg 4276	. . 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 4233	. 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 2202   {cab 2217   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   X. cxp 4729   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  fnmap  6867  mapvalg  6870  exmidpw2en  7147  nninfex  7363  ptex  13410  isghm  13893  psrval  14745  psrbasg  14758  cnovex  14990  ispsmet  15117  cncfval  15366  wksfval  16246  wlkex  16249