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Theorem cnvi 6161
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 3484 . . . . 5 𝑥 ∈ V
21ideq 5863 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2017 . . . 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 5693 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5578 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2775 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540   class class class wbr 5143  {copab 5205   I cid 5577  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  coi2  6283  funi  6598  cnvresid  6645  fcoi1  6782  ssdomg  9040  mbfid  25670  mthmpps  35587  brid  38307  extid  38311  cosscnvid  38482  idsymrel  38562
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