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Theorem cnvi 5262
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 3060 . . . . 5  |-  x  e. 
_V
21ideq 5009 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1873 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 257 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 4483 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4864 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4771 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2494 1  |-  `'  _I  =  _I
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   class class class wbr 4418   {copab 4476    _I cid 4766   `'ccnv 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864
This theorem is referenced by:  coi2  5375  funi  5635  cnvresid  5679  fcoi1  5784  ssdomg  7646  mbfid  22648  mthmpps  30270
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