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Theorem cnvi 5013
Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvi  |-  `'  _I  =  _I

Proof of Theorem cnvi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2733 . . . . 5  |-  x  e. 
_V
21ideq 4761 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1699 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 183 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 4054 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4617 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4276 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2201 1  |-  `'  _I  =  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1348   class class class wbr 3987   {copab 4047    _I cid 4271   `'ccnv 4608
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-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617
This theorem is referenced by:  coi2  5125  funi  5228  cnvresid  5270  fcoi1  5376  ssdomg  6754
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