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Theorem mapex 6987
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 5565 . . . 4  |-  ( f : A --> B  -> 
f  C_  ( A  X.  B ) )
21ss2abi 3379 . . 3  |-  { f  |  f : A --> B }  C_  { f  |  f  C_  ( A  X.  B ) }
3 df-pw 3765 . . 3  |-  ~P ( A  X.  B )  =  { f  |  f 
C_  ( A  X.  B ) }
42, 3sseqtr4i 3345 . 2  |-  { f  |  f : A --> B }  C_  ~P ( A  X.  B )
5 xpexg 4952 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
6 pwexg 4347 . . 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 4313 . 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 2394   _Vcvv 2920    C_ wss 3284   ~Pcpw 3763    X. cxp 4839   -->wf 5413
This theorem is referenced by:  fnmap  6988  mapvalg  6991  isghm  14965  wlks  21483  wlkres  21486  trls  21493  crcts  21566  cycls  21567  measbase  24508  measval  24509  ismeas  24510  isrnmeas  24511  cnfex  27570
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-cnv 4849  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421
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