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Theorem mapex 6983
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 5561 . . . 4  |-  ( f : A --> B  -> 
f  C_  ( A  X.  B ) )
21ss2abi 3375 . . 3  |-  { f  |  f : A --> B }  C_  { f  |  f  C_  ( A  X.  B ) }
3 df-pw 3761 . . 3  |-  ~P ( A  X.  B )  =  { f  |  f 
C_  ( A  X.  B ) }
42, 3sseqtr4i 3341 . 2  |-  { f  |  f : A --> B }  C_  ~P ( A  X.  B )
5 xpexg 4948 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
6 pwexg 4343 . . 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 4309 . 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 2390   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759    X. cxp 4835   -->wf 5409
This theorem is referenced by:  fnmap  6984  mapvalg  6987  isghm  14961  wlks  21479  wlkres  21482  trls  21489  crcts  21562  cycls  21563  measbase  24504  measval  24505  ismeas  24506  isrnmeas  24507  cnfex  27566
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417
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