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Theorem f1oabexg 5645
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 5633 . . . . 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 3375 . . 3  |-  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }
5 eqid 2404 . . . 4  |-  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  |  ( f : A --> B  /\  ph ) }
65fabexg 5583 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )
7 ssexg 4309 . . 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 2488 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 1721   {cab 2390   _Vcvv 2916    C_ wss 3280   -->wf 5409   -1-1-onto->wf1o 5412
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-f1o 5420
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