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Theorem funcnvsn 5379
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5382 via cnvsn 5237, 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 5133 . 2  |-  Rel  `' { <. A ,  B >. }
2 moeq 3017 . . . 4  |-  E* y 
y  =  A
3 vex 2867 . . . . . . . 8  |-  x  e. 
_V
4 vex 2867 . . . . . . . 8  |-  y  e. 
_V
53, 4brcnv 4946 . . . . . . 7  |-  ( x `' { <. A ,  B >. } y  <->  y { <. A ,  B >. } x )
6 df-br 4105 . . . . . . 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 3740 . . . . . . 7  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  <. y ,  x >.  =  <. A ,  B >. )
94, 3opth1 4326 . . . . . . 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 2256 . . . 4  |-  ( E* y  y  =  A  ->  E* y  x `' { <. A ,  B >. } y )
132, 12ax-mp 8 . . 3  |-  E* y  x `' { <. A ,  B >. } y
1413ax-gen 1546 . 2  |-  A. x E* y  x `' { <. A ,  B >. } y
15 dffun6 5352 . 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 1540    = wceq 1642    e. wcel 1710   E*wmo 2210   {csn 3716   <.cop 3719   class class class wbr 4104   `'ccnv 4770   Rel wrel 4776   Fun wfun 5331
This theorem is referenced by:  funsng  5380  strlemor1  13332  0spth  27713  2pthonlem1  27735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-fun 5339
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