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Theorem ercnv 6530
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 6518 . 2  |-  ( R  Er  A  ->  Rel  R )
2 relcnv 4987 . . 3  |-  Rel  `' R
3 id 19 . . . . . 6  |-  ( R  Er  A  ->  R  Er  A )
43ersymb 6523 . . . . 5  |-  ( R  Er  A  ->  (
y R x  <->  x R
y ) )
5 vex 2733 . . . . . . 7  |-  x  e. 
_V
6 vex 2733 . . . . . . 7  |-  y  e. 
_V
75, 6brcnv 4792 . . . . . 6  |-  ( x `' R y  <->  y R x )
8 df-br 3988 . . . . . 6  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
97, 8bitr3i 185 . . . . 5  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
10 df-br 3988 . . . . 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 4708 . . 3  |-  ( ( ( Rel  `' R  /\  Rel  R )  /\  R  Er  A )  ->  `' R  =  R
)
132, 12mpanl1 432 . 2  |-  ( ( Rel  R  /\  R  Er  A )  ->  `' R  =  R )
141, 13mpancom 420 1  |-  ( R  Er  A  ->  `' R  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   <.cop 3584   class class class wbr 3987   `'ccnv 4608   Rel wrel 4614    Er wer 6506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-er 6509
This theorem is referenced by:  errn  6531
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