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Theorem funcnvsn 5499
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5502 via cnvsn 5355, 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 5245 . 2  |-  Rel  `' { <. A ,  B >. }
2 moeq 3112 . . . 4  |-  E* y 
y  =  A
3 vex 2961 . . . . . . . 8  |-  x  e. 
_V
4 vex 2961 . . . . . . . 8  |-  y  e. 
_V
53, 4brcnv 5058 . . . . . . 7  |-  ( x `' { <. A ,  B >. } y  <->  y { <. A ,  B >. } x )
6 df-br 4216 . . . . . . 7  |-  ( y { <. A ,  B >. } x  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
75, 6bitri 242 . . . . . 6  |-  ( x `' { <. A ,  B >. } y  <->  <. y ,  x >.  e.  { <. A ,  B >. } )
8 elsni 3840 . . . . . . 7  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  <. y ,  x >.  =  <. A ,  B >. )
94, 3opth1 4437 . . . . . . 7  |-  ( <.
y ,  x >.  = 
<. A ,  B >.  -> 
y  =  A )
108, 9syl 16 . . . . . 6  |-  ( <.
y ,  x >.  e. 
{ <. A ,  B >. }  ->  y  =  A )
117, 10sylbi 189 . . . . 5  |-  ( x `' { <. A ,  B >. } y  ->  y  =  A )
1211moimi 2330 . . . 4  |-  ( E* y  y  =  A  ->  E* y  x `' { <. A ,  B >. } y )
132, 12ax-mp 5 . . 3  |-  E* y  x `' { <. A ,  B >. } y
1413ax-gen 1556 . 2  |-  A. x E* y  x `' { <. A ,  B >. } y
15 dffun6 5472 . 2  |-  ( Fun  `' { <. A ,  B >. }  <->  ( Rel  `' { <. A ,  B >. }  /\  A. x E* y  x `' { <. A ,  B >. } y ) )
161, 14, 15mpbir2an 888 1  |-  Fun  `' { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:   A.wal 1550    = wceq 1653    e. wcel 1726   E*wmo 2284   {csn 3816   <.cop 3819   class class class wbr 4215   `'ccnv 4880   Rel wrel 4886   Fun wfun 5451
This theorem is referenced by:  funsng  5500  strlemor1  13561  0spth  21576  2pthlem1  21600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-fun 5459
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