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Theorem f1oabexg 5500
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 5488 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
32anim1i 551 . . . 4  |-  ( ( f : A -1-1-onto-> B  /\  ph )  ->  ( f : A --> B  /\  ph ) )
43ss2abi 3258 . . 3  |-  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  C_  { f  |  ( f : A --> B  /\  ph ) }
5 eqid 2296 . . . 4  |-  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  |  ( f : A --> B  /\  ph ) }
65fabexg 5438 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  e.  _V )
7 ssexg 4176 . . 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 644 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }  e.  _V )
91, 8syl5eqel 2380 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   -->wf 5267   -1-1-onto->wf1o 5270
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  df-f1 5276  df-f1o 5278
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