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Theorem funcnvsn 5455
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5458 via cnvsn 5311, 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 5201 . 2  |-  Rel  `' { <. A ,  B >. }
2 moeq 3070 . . . 4  |-  E* y 
y  =  A
3 vex 2919 . . . . . . . 8  |-  x  e. 
_V
4 vex 2919 . . . . . . . 8  |-  y  e. 
_V
53, 4brcnv 5014 . . . . . . 7  |-  ( x `' { <. A ,  B >. } y  <->  y { <. A ,  B >. } x )
6 df-br 4173 . . . . . . 7  |-  ( y { <. A ,  B >. } x  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
75, 6bitri 241 . . . . . 6  |-  ( x `' { <. A ,  B >. } y  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
8 elsni 3798 . . . . . . 7  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  <. y ,  x >.  =  <. A ,  B >. )
94, 3opth1 4394 . . . . . . 7  |-  ( <.
y ,  x >.  = 
<. A ,  B >.  -> 
y  =  A )
108, 9syl 16 . . . . . 6  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  y  =  A )
117, 10sylbi 188 . . . . 5  |-  ( x `' { <. A ,  B >. } y  ->  y  =  A )
1211moimi 2301 . . . 4  |-  ( E* y  y  =  A  ->  E* y  x `' { <. A ,  B >. } y )
132, 12ax-mp 8 . . 3  |-  E* y  x `' { <. A ,  B >. } y
1413ax-gen 1552 . 2  |-  A. x E* y  x `' { <. A ,  B >. } y
15 dffun6 5428 . 2  |-  ( Fun  `' { <. A ,  B >. }  <->  ( Rel  `' { <. A ,  B >. }  /\  A. x E* y  x `' { <. A ,  B >. } y ) )
161, 14, 15mpbir2an 887 1  |-  Fun  `' { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:   A.wal 1546    = wceq 1649    e. wcel 1721   E*wmo 2255   {csn 3774   <.cop 3777   class class class wbr 4172   `'ccnv 4836   Rel wrel 4842   Fun wfun 5407
This theorem is referenced by:  funsng  5456  strlemor1  13511  0spth  21524  2pthlem1  21548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-fun 5415
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