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Theorem fabexg 6730
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 6702 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 pwexg 4624 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
3 fabexg.1 . . . . 5  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
4 fssxp 5734 . . . . . . . 8  |-  ( x : A --> B  ->  x  C_  ( A  X.  B ) )
5 selpw 4010 . . . . . . . 8  |-  ( x  e.  ~P ( A  X.  B )  <->  x  C_  ( A  X.  B ) )
64, 5sylibr 212 . . . . . . 7  |-  ( x : A --> B  ->  x  e.  ~P ( A  X.  B ) )
76anim1i 568 . . . . . 6  |-  ( ( x : A --> B  /\  ph )  ->  ( x  e.  ~P ( A  X.  B )  /\  ph ) )
87ss2abi 3565 . . . . 5  |-  { x  |  ( x : A --> B  /\  ph ) }  C_  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }
93, 8eqsstri 3527 . . . 4  |-  F  C_  { x  |  ( x  e.  ~P ( A  X.  B )  /\  ph ) }
10 ssab2 3577 . . . 4  |-  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }  C_  ~P ( A  X.  B )
119, 10sstri 3506 . . 3  |-  F  C_  ~P ( A  X.  B
)
12 ssexg 4586 . . 3  |-  ( ( F  C_  ~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
1311, 12mpan 670 . 2  |-  ( ~P ( A  X.  B
)  e.  _V  ->  F  e.  _V )
141, 2, 133syl 20 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2445   _Vcvv 3106    C_ wss 3469   ~Pcpw 4003    X. cxp 4990   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583
This theorem is referenced by:  fabex  6731  f1oabexg  6733  elghomlem1  25025
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