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Theorem cnvi 6141
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 3474 . . . . 5 𝑥 ∈ V
21ideq 5850 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2014 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 275 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5681 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5571 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2766 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534   class class class wbr 5143  {copab 5205   I cid 5570  ccnv 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681
This theorem is referenced by:  coi2  6262  funi  6580  cnvresid  6627  fcoi1  6766  ssdomg  9015  mbfid  25558  mthmpps  35187  brid  37773  extid  37777  cosscnvid  37948  idsymrel  38028
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