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Theorem mapex 6732
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 5324 . . . 4  |-  ( f : A --> B  -> 
f  C_  ( A  X.  B ) )
21ss2abi 3206 . . 3  |-  { f  |  f : A --> B }  C_  { f  |  f  C_  ( A  X.  B ) }
3 df-pw 3587 . . 3  |-  ~P ( A  X.  B )  =  { f  |  f 
C_  ( A  X.  B ) }
42, 3sseqtr4i 3172 . 2  |-  { f  |  f : A --> B }  C_  ~P ( A  X.  B )
5 xpexg 4774 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
6 pwexg 4152 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
75, 6syl 17 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( A  X.  B )  e.  _V )
8 ssexg 4120 . 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 647 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : A --> B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   {cab 2242   _Vcvv 2757    C_ wss 3113   ~Pcpw 3585    X. cxp 4645   -->wf 4655
This theorem is referenced by:  fnmap  6733  mapvalg  6736  isghm  14631  cnfex  27053
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-rel 4662  df-cnv 4663  df-dm 4665  df-rn 4666  df-fun 4669  df-fn 4670  df-f 4671
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