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Theorem fabexg 5438
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 4816 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 pwexg 4210 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ~P ( A  X.  B
)  e.  _V )
3 fabexg.1 . . . . 5  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
4 fssxp 5416 . . . . . . . 8  |-  ( x : A --> B  ->  x  C_  ( A  X.  B ) )
5 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
65elpw 3644 . . . . . . . 8  |-  ( x  e.  ~P ( A  X.  B )  <->  x  C_  ( A  X.  B ) )
74, 6sylibr 203 . . . . . . 7  |-  ( x : A --> B  ->  x  e.  ~P ( A  X.  B ) )
87anim1i 551 . . . . . 6  |-  ( ( x : A --> B  /\  ph )  ->  ( x  e.  ~P ( A  X.  B )  /\  ph ) )
98ss2abi 3258 . . . . 5  |-  { x  |  ( x : A --> B  /\  ph ) }  C_  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }
103, 9eqsstri 3221 . . . 4  |-  F  C_  { x  |  ( x  e.  ~P ( A  X.  B )  /\  ph ) }
11 ssab2 3270 . . . 4  |-  { x  |  ( x  e. 
~P ( A  X.  B )  /\  ph ) }  C_  ~P ( A  X.  B )
1210, 11sstri 3201 . . 3  |-  F  C_  ~P ( A  X.  B
)
13 ssexg 4176 . . 3  |-  ( ( F  C_  ~P ( A  X.  B )  /\  ~P ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
1412, 13mpan 651 . 2  |-  ( ~P ( A  X.  B
)  e.  _V  ->  F  e.  _V )
151, 2, 143syl 18 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638    X. cxp 4703   -->wf 5267
This theorem is referenced by:  fabex  5439  f1oabexg  5500  elghomlem1  21044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275
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