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Theorem ercnv 6063
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
ercnv  R  Er  `' R  R

Proof of Theorem ercnv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 errel 6051 . 2  R  Er  Rel  R
2 relcnv 4646 . . 3  Rel  `' R
3 id 19 . . . . . 6  R  Er  R  Er
43ersymb 6056 . . . . 5  R  Er  R  R
5 vex 2554 . . . . . . 7 
_V
6 vex 2554 . . . . . . 7 
_V
75, 6brcnv 4461 . . . . . 6  `' R  R
8 df-br 3756 . . . . . 6  `' R  <. ,  >.  `' R
97, 8bitr3i 175 . . . . 5  R  <. ,  >.  `' R
10 df-br 3756 . . . . 5  R  <. ,  >.  R
114, 9, 103bitr3g 211 . . . 4  R  Er  <. ,  >.  `' R  <. ,  >.  R
1211eqrelrdv2 4382 . . 3  Rel  `' R  Rel  R  R  Er  `' R  R
132, 12mpanl1 410 . 2  Rel  R  R  Er  `' R  R
141, 13mpancom 399 1  R  Er  `' R  R
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   <.cop 3370   class class class wbr 3755   `'ccnv 4287   Rel wrel 4293    Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-er 6042
This theorem is referenced by:  errn  6064
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