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Theorem cnvi 5085
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem cnvi
StepHypRef Expression
1 vex 2793 . . . . 5  |-  x  e. 
_V
21ideq 4836 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1648 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 242 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 4085 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4697 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4309 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2315 1  |-  `'  _I  =  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1624   class class class wbr 4025   {copab 4078    _I cid 4304   `'ccnv 4688
This theorem is referenced by:  coi2  5188  funi  5251  cnvresid  5288  fcoi1  5381  ssdomg  6903  mbfid  18986
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697
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