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Theorem f1oabexg 5265
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 5253 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
32anim1i 333 . . . 4  |-  ( ( f : A -1-1-onto-> B  /\  ph )  ->  ( f : A --> B  /\  ph ) )
43ss2abi 3093 . . 3  |-  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }
5 eqid 2088 . . . 4  |-  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  |  ( f : A --> B  /\  ph ) }
65fabexg 5198 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )
7 ssexg 3978 . . 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 405 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  e.  _V )
91, 8syl5eqel 2174 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619    C_ wss 2999   -->wf 5011   -1-1-onto->wf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-f1o 5022
This theorem is referenced by: (None)
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