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Theorem mapex 6991
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
StepHypRef Expression
1 fssxp 5569 . . . 4  |-  ( f : A --> B  -> 
f  C_  ( A  X.  B ) )
21ss2abi 3383 . . 3  |-  { f  |  f : A --> B }  C_  { f  |  f  C_  ( A  X.  B ) }
3 df-pw 3769 . . 3  |-  ~P ( A  X.  B )  =  { f  |  f 
C_  ( A  X.  B ) }
42, 3sseqtr4i 3349 . 2  |-  { f  |  f : A --> B }  C_  ~P ( A  X.  B )
5 xpexg 4956 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
6 pwexg 4351 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
75, 6syl 16 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( A  X.  B )  e.  _V )
8 ssexg 4317 . 2  |-  ( ( { f  |  f : A --> B }  C_ 
~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  { f  |  f : A --> B }  e.  _V )
94, 7, 8sylancr 645 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : A --> B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   {cab 2398   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767    X. cxp 4843   -->wf 5417
This theorem is referenced by:  fnmap  6992  mapvalg  6995  isghm  14969  wlks  21487  wlkres  21490  trls  21497  crcts  21570  cycls  21571  measbase  24512  measval  24513  ismeas  24514  isrnmeas  24515  cnfex  27574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-fun 5423  df-fn 5424  df-f 5425
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