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Theorem cnvi 5227
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 3061 . . . . 5  |-  x  e. 
_V
21ideq 4975 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1818 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 249 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 4458 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4830 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4737 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2441 1  |-  `'  _I  =  _I
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   class class class wbr 4394   {copab 4451    _I cid 4732   `'ccnv 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830
This theorem is referenced by:  coi2  5339  funi  5598  cnvresid  5638  fcoi1  5741  ssdomg  7598  mbfid  22333  mthmpps  29781
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