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Theorem ercnv 6327
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
ercnv  |-  ( R  Er  A  ->  `' R  =  R )

Proof of Theorem ercnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 errel 6315 . 2  |-  ( R  Er  A  ->  Rel  R )
2 relcnv 4823 . . 3  |-  Rel  `' R
3 id 19 . . . . . 6  |-  ( R  Er  A  ->  R  Er  A )
43ersymb 6320 . . . . 5  |-  ( R  Er  A  ->  (
y R x  <->  x R
y ) )
5 vex 2623 . . . . . . 7  |-  x  e. 
_V
6 vex 2623 . . . . . . 7  |-  y  e. 
_V
75, 6brcnv 4632 . . . . . 6  |-  ( x `' R y  <->  y R x )
8 df-br 3852 . . . . . 6  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
97, 8bitr3i 185 . . . . 5  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
10 df-br 3852 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
114, 9, 103bitr3g 221 . . . 4  |-  ( R  Er  A  ->  ( <. x ,  y >.  e.  `' R  <->  <. x ,  y
>.  e.  R ) )
1211eqrelrdv2 4550 . . 3  |-  ( ( ( Rel  `' R  /\  Rel  R )  /\  R  Er  A )  ->  `' R  =  R
)
132, 12mpanl1 426 . 2  |-  ( ( Rel  R  /\  R  Er  A )  ->  `' R  =  R )
141, 13mpancom 414 1  |-  ( R  Er  A  ->  `' R  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439   <.cop 3453   class class class wbr 3851   `'ccnv 4451   Rel wrel 4457    Er wer 6303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-er 6306
This theorem is referenced by:  errn  6328
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