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Theorem ercnv 6450
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 6438 . 2  |-  ( R  Er  A  ->  Rel  R )
2 relcnv 4917 . . 3  |-  Rel  `' R
3 id 19 . . . . . 6  |-  ( R  Er  A  ->  R  Er  A )
43ersymb 6443 . . . . 5  |-  ( R  Er  A  ->  (
y R x  <->  x R
y ) )
5 vex 2689 . . . . . . 7  |-  x  e. 
_V
6 vex 2689 . . . . . . 7  |-  y  e. 
_V
75, 6brcnv 4722 . . . . . 6  |-  ( x `' R y  <->  y R x )
8 df-br 3930 . . . . . 6  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
97, 8bitr3i 185 . . . . 5  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
10 df-br 3930 . . . . 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 4638 . . 3  |-  ( ( ( Rel  `' R  /\  Rel  R )  /\  R  Er  A )  ->  `' R  =  R
)
132, 12mpanl1 430 . 2  |-  ( ( Rel  R  /\  R  Er  A )  ->  `' R  =  R )
141, 13mpancom 418 1  |-  ( R  Er  A  ->  `' R  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   <.cop 3530   class class class wbr 3929   `'ccnv 4538   Rel wrel 4544    Er wer 6426
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-er 6429
This theorem is referenced by:  errn  6451
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