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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
StepHypRef Expression
1 fssxp 5483 . . . 4  |-  ( f : A --> B  -> 
f  C_  ( A  X.  B ) )
21ss2abi 3321 . . 3  |-  { f  |  f : A --> B }  C_  { f  |  f  C_  ( A  X.  B ) }
3 df-pw 3703 . . 3  |-  ~P ( A  X.  B )  =  { f  |  f 
C_  ( A  X.  B ) }
42, 3sseqtr4i 3287 . 2  |-  { f  |  f : A --> B }  C_  ~P ( A  X.  B )
5 xpexg 4882 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
6 pwexg 4275 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
75, 6syl 15 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( A  X.  B )  e.  _V )
8 ssexg 4241 . 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 644 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : A --> B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   {cab 2344   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701    X. cxp 4769   -->wf 5333
This theorem is referenced by:  fnmap  6867  mapvalg  6870  isghm  14782  measbase  23817  measval  23818  ismeas  23819  isrnmeas  23820  cnfex  27022  wlks  27678  wlkres  27681  trls  27688  crcts  27745  cycls  27746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rn 4782  df-fun 5339  df-fn 5340  df-f 5341
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