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Theorem ercnv 6929
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 6917 . 2  |-  ( R  Er  A  ->  Rel  R )
2 relcnv 5245 . . 3  |-  Rel  `' R
3 id 21 . . . . . 6  |-  ( R  Er  A  ->  R  Er  A )
43ersymb 6922 . . . . 5  |-  ( R  Er  A  ->  (
y R x  <->  x R
y ) )
5 vex 2961 . . . . . . 7  |-  x  e. 
_V
6 vex 2961 . . . . . . 7  |-  y  e. 
_V
75, 6brcnv 5058 . . . . . 6  |-  ( x `' R y  <->  y R x )
8 df-br 4216 . . . . . 6  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
97, 8bitr3i 244 . . . . 5  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
10 df-br 4216 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
114, 9, 103bitr3g 280 . . . 4  |-  ( R  Er  A  ->  ( <. x ,  y >.  e.  `' R  <->  <. x ,  y
>.  e.  R ) )
1211eqrelrdv2 4978 . . 3  |-  ( ( ( Rel  `' R  /\  Rel  R )  /\  R  Er  A )  ->  `' R  =  R
)
132, 12mpanl1 663 . 2  |-  ( ( Rel  R  /\  R  Er  A )  ->  `' R  =  R )
141, 13mpancom 652 1  |-  ( R  Er  A  ->  `' R  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215   `'ccnv 4880   Rel wrel 4886    Er wer 6905
This theorem is referenced by:  errn  6930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-er 6908
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