ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ercnv Unicode version

Theorem ercnv 6570
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 6558 . 2  |-  ( R  Er  A  ->  Rel  R )
2 relcnv 5018 . . 3  |-  Rel  `' R
3 id 19 . . . . . 6  |-  ( R  Er  A  ->  R  Er  A )
43ersymb 6563 . . . . 5  |-  ( R  Er  A  ->  (
y R x  <->  x R
y ) )
5 vex 2752 . . . . . . 7  |-  x  e. 
_V
6 vex 2752 . . . . . . 7  |-  y  e. 
_V
75, 6brcnv 4822 . . . . . 6  |-  ( x `' R y  <->  y R x )
8 df-br 4016 . . . . . 6  |-  ( x `' R y  <->  <. x ,  y >.  e.  `' R )
97, 8bitr3i 186 . . . . 5  |-  ( y R x  <->  <. x ,  y >.  e.  `' R )
10 df-br 4016 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
114, 9, 103bitr3g 222 . . . 4  |-  ( R  Er  A  ->  ( <. x ,  y >.  e.  `' R  <->  <. x ,  y
>.  e.  R ) )
1211eqrelrdv2 4737 . . 3  |-  ( ( ( Rel  `' R  /\  Rel  R )  /\  R  Er  A )  ->  `' R  =  R
)
132, 12mpanl1 434 . 2  |-  ( ( Rel  R  /\  R  Er  A )  ->  `' R  =  R )
141, 13mpancom 422 1  |-  ( R  Er  A  ->  `' R  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   <.cop 3607   class class class wbr 4015   `'ccnv 4637   Rel wrel 4643    Er wer 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-er 6549
This theorem is referenced by:  errn  6571
  Copyright terms: Public domain W3C validator