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Theorem funcnvsn 5297
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5300 via cnvsn 5155, but stating it this way allows us to skip the sethood assumptions on  A and  B. (Contributed by NM, 30-Apr-2015.)
Assertion
Ref Expression
funcnvsn  |-  Fun  `' { <. A ,  B >. }

Proof of Theorem funcnvsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5051 . 2  |-  Rel  `' { <. A ,  B >. }
2 moeq 2941 . . . 4  |-  E* y 
y  =  A
3 vex 2791 . . . . . . . 8  |-  x  e. 
_V
4 vex 2791 . . . . . . . 8  |-  y  e. 
_V
53, 4brcnv 4864 . . . . . . 7  |-  ( x `' { <. A ,  B >. } y  <->  y { <. A ,  B >. } x )
6 df-br 4024 . . . . . . 7  |-  ( y { <. A ,  B >. } x  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
75, 6bitri 240 . . . . . 6  |-  ( x `' { <. A ,  B >. } y  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
8 elsni 3664 . . . . . . 7  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  <. y ,  x >.  =  <. A ,  B >. )
94, 3opth1 4244 . . . . . . 7  |-  ( <.
y ,  x >.  = 
<. A ,  B >.  -> 
y  =  A )
108, 9syl 15 . . . . . 6  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  y  =  A )
117, 10sylbi 187 . . . . 5  |-  ( x `' { <. A ,  B >. } y  ->  y  =  A )
1211moimi 2190 . . . 4  |-  ( E* y  y  =  A  ->  E* y  x `' { <. A ,  B >. } y )
132, 12ax-mp 8 . . 3  |-  E* y  x `' { <. A ,  B >. } y
1413ax-gen 1533 . 2  |-  A. x E* y  x `' { <. A ,  B >. } y
15 dffun6 5270 . 2  |-  ( Fun  `' { <. A ,  B >. }  <->  ( Rel  `' { <. A ,  B >. }  /\  A. x E* y  x `' { <. A ,  B >. } y ) )
161, 14, 15mpbir2an 886 1  |-  Fun  `' { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144   {csn 3640   <.cop 3643   class class class wbr 4023   `'ccnv 4688   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  funsng  5298  strlemor1  13235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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