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Theorem cnvi 6132
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 3470 . . . . 5 𝑥 ∈ V
21ideq 5843 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2013 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 275 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5675 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5565 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2762 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   class class class wbr 5139  {copab 5201   I cid 5564  ccnv 5666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675
This theorem is referenced by:  coi2  6253  funi  6571  cnvresid  6618  fcoi1  6756  ssdomg  8993  mbfid  25488  mthmpps  35064  brid  37669  extid  37673  cosscnvid  37845  idsymrel  37925
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