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Theorem fabexg 5624
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 4989 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 pwexg 4383 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
3 fabexg.1 . . . . 5  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
4 fssxp 5602 . . . . . . . 8  |-  ( x : A --> B  ->  x  C_  ( A  X.  B ) )
5 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
65elpw 3805 . . . . . . . 8  |-  ( x  e.  ~P ( A  X.  B )  <->  x  C_  ( A  X.  B ) )
74, 6sylibr 204 . . . . . . 7  |-  ( x : A --> B  ->  x  e.  ~P ( A  X.  B ) )
87anim1i 552 . . . . . 6  |-  ( ( x : A --> B  /\  ph )  ->  ( x  e.  ~P ( A  X.  B )  /\  ph ) )
98ss2abi 3415 . . . . 5  |-  { x  |  ( x : A --> B  /\  ph ) }  C_  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }
103, 9eqsstri 3378 . . . 4  |-  F  C_  { x  |  ( x  e.  ~P ( A  X.  B )  /\  ph ) }
11 ssab2 3427 . . . 4  |-  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }  C_  ~P ( A  X.  B )
1210, 11sstri 3357 . . 3  |-  F  C_  ~P ( A  X.  B
)
13 ssexg 4349 . . 3  |-  ( ( F  C_  ~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
1412, 13mpan 652 . 2  |-  ( ~P ( A  X.  B
)  e.  _V  ->  F  e.  _V )
151, 2, 143syl 19 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799    X. cxp 4876   -->wf 5450
This theorem is referenced by:  fabex  5625  f1oabexg  5686  elghomlem1  21949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458
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