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Theorem f1oabexg 5678
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 5666 . . . . 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 3407 . . 3  |-  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }
5 eqid 2435 . . . 4  |-  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  |  ( f : A --> B  /\  ph ) }
65fabexg 5616 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )
7 ssexg 4341 . . 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 2519 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    C_ wss 3312   -->wf 5442   -1-1-onto->wf1o 5445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-f1o 5453
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