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Theorem cnvi 4823
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 2622 . . . . 5  |-  x  e. 
_V
21ideq 4576 . . . 4  |-  ( y  _I  x  <->  y  =  x )
3 equcom 1639 . . . 4  |-  ( y  =  x  <->  x  =  y )
42, 3bitri 182 . . 3  |-  ( y  _I  x  <->  x  =  y )
54opabbii 3897 . 2  |-  { <. x ,  y >.  |  y  _I  x }  =  { <. x ,  y
>.  |  x  =  y }
6 df-cnv 4436 . 2  |-  `'  _I  =  { <. x ,  y
>.  |  y  _I  x }
7 df-id 4111 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
85, 6, 73eqtr4i 2118 1  |-  `'  _I  =  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1289   class class class wbr 3837   {copab 3890    _I cid 4106   `'ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  coi2  4934  funi  5032  cnvresid  5074  fcoi1  5175  ssdomg  6475
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