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Theorem fabexg 5192
Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypothesis
Ref Expression
fabexg.1  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
Assertion
Ref Expression
fabexg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)    D( x)    F( x)

Proof of Theorem fabexg
StepHypRef Expression
1 xpexg 4548 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 pwexg 4013 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
3 fabexg.1 . . . . 5  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
4 fssxp 5172 . . . . . . . 8  |-  ( x : A --> B  ->  x  C_  ( A  X.  B ) )
5 selpw 3434 . . . . . . . 8  |-  ( x  e.  ~P ( A  X.  B )  <->  x  C_  ( A  X.  B ) )
64, 5sylibr 132 . . . . . . 7  |-  ( x : A --> B  ->  x  e.  ~P ( A  X.  B ) )
76anim1i 333 . . . . . 6  |-  ( ( x : A --> B  /\  ph )  ->  ( x  e.  ~P ( A  X.  B )  /\  ph ) )
87ss2abi 3093 . . . . 5  |-  { x  |  ( x : A --> B  /\  ph ) }  C_  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }
93, 8eqsstri 3056 . . . 4  |-  F  C_  { x  |  ( x  e.  ~P ( A  X.  B )  /\  ph ) }
10 ssab2 3105 . . . 4  |-  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }  C_  ~P ( A  X.  B )
119, 10sstri 3034 . . 3  |-  F  C_  ~P ( A  X.  B
)
12 ssexg 3976 . . 3  |-  ( ( F  C_  ~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
1311, 12mpan 415 . 2  |-  ( ~P ( A  X.  B
)  e.  _V  ->  F  e.  _V )
141, 2, 133syl 17 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619    C_ wss 2999   ~Pcpw 3427    X. cxp 4434   -->wf 5006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-dm 4446  df-rn 4447  df-fun 5012  df-fn 5013  df-f 5014
This theorem is referenced by:  fabex  5193  f1oabexg  5259
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