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Theorem f1oabexg 5626
Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypothesis
Ref Expression
f1oabexg.1  |-  F  =  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }
Assertion
Ref Expression
f1oabexg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Distinct variable groups:    A, f    B, f
Allowed substitution hints:    ph( f)    C( f)    D( f)    F( f)

Proof of Theorem f1oabexg
StepHypRef Expression
1 f1oabexg.1 . 2  |-  F  =  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }
2 f1of 5614 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
32anim1i 552 . . . 4  |-  ( ( f : A -1-1-onto-> B  /\  ph )  ->  ( f : A --> B  /\  ph ) )
43ss2abi 3358 . . 3  |-  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }
5 eqid 2387 . . . 4  |-  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  |  ( f : A --> B  /\  ph ) }
65fabexg 5564 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )
7 ssexg 4290 . . 3  |-  ( ( { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }  /\  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )  ->  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  e.  _V )
84, 6, 7sylancr 645 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  e.  _V )
91, 8syl5eqel 2471 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899    C_ wss 3263   -->wf 5390   -1-1-onto->wf1o 5393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-dm 4828  df-rn 4829  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-f1o 5401
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