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Theorem fabexg 6532
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 6506 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 pwexg 4473 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
3 fabexg.1 . . . . 5  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
4 fssxp 5567 . . . . . . . 8  |-  ( x : A --> B  ->  x  C_  ( A  X.  B ) )
5 selpw 3864 . . . . . . . 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 565 . . . . . 6  |-  ( ( x : A --> B  /\  ph )  ->  ( x  e.  ~P ( A  X.  B )  /\  ph ) )
87ss2abi 3421 . . . . 5  |-  { x  |  ( x : A --> B  /\  ph ) }  C_  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }
93, 8eqsstri 3383 . . . 4  |-  F  C_  { x  |  ( x  e.  ~P ( A  X.  B )  /\  ph ) }
10 ssab2 3433 . . . 4  |-  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }  C_  ~P ( A  X.  B )
119, 10sstri 3362 . . 3  |-  F  C_  ~P ( A  X.  B
)
12 ssexg 4435 . . 3  |-  ( ( F  C_  ~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
1311, 12mpan 665 . 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 1364    e. wcel 1761   {cab 2427   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857    X. cxp 4834   -->wf 5411
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-fun 5417  df-fn 5418  df-f 5419
This theorem is referenced by:  fabex  6533  f1oabexg  6535  elghomlem1  23783
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