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Theorem cnvsng 5237
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
Assertion
Ref Expression
cnvsng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )

Proof of Theorem cnvsng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3875 . . . . 5  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21sneqd 3729 . . . 4  |-  ( x  =  A  ->  { <. x ,  y >. }  =  { <. A ,  y
>. } )
32cnveqd 4936 . . 3  |-  ( x  =  A  ->  `' { <. x ,  y
>. }  =  `' { <. A ,  y >. } )
4 opeq2 3876 . . . 4  |-  ( x  =  A  ->  <. y ,  x >.  =  <. y ,  A >. )
54sneqd 3729 . . 3  |-  ( x  =  A  ->  { <. y ,  x >. }  =  { <. y ,  A >. } )
63, 5eqeq12d 2372 . 2  |-  ( x  =  A  ->  ( `' { <. x ,  y
>. }  =  { <. y ,  x >. }  <->  `' { <. A ,  y >. }  =  { <. y ,  A >. } ) )
7 opeq2 3876 . . . . 5  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
87sneqd 3729 . . . 4  |-  ( y  =  B  ->  { <. A ,  y >. }  =  { <. A ,  B >. } )
98cnveqd 4936 . . 3  |-  ( y  =  B  ->  `' { <. A ,  y
>. }  =  `' { <. A ,  B >. } )
10 opeq1 3875 . . . 4  |-  ( y  =  B  ->  <. y ,  A >.  =  <. B ,  A >. )
1110sneqd 3729 . . 3  |-  ( y  =  B  ->  { <. y ,  A >. }  =  { <. B ,  A >. } )
129, 11eqeq12d 2372 . 2  |-  ( y  =  B  ->  ( `' { <. A ,  y
>. }  =  { <. y ,  A >. }  <->  `' { <. A ,  B >. }  =  { <. B ,  A >. } ) )
13 vex 2867 . . 3  |-  x  e. 
_V
14 vex 2867 . . 3  |-  y  e. 
_V
1513, 14cnvsn 5234 . 2  |-  `' { <. x ,  y >. }  =  { <. y ,  x >. }
166, 12, 15vtocl2g 2923 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {csn 3716   <.cop 3719   `'ccnv 4767
This theorem is referenced by:  opswap  5238  funsng  5377  f1oprswap  5595  constr3pthlem2  27763
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 4220  ax-nul 4228  ax-pr 4293
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 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776
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