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Theorem cnvi 6138
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 3479 . . . . 5 𝑥 ∈ V
21ideq 5850 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2022 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 275 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5214 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5683 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5573 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2771 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   class class class wbr 5147  {copab 5209   I cid 5572  ccnv 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683
This theorem is referenced by:  coi2  6259  funi  6577  cnvresid  6624  fcoi1  6762  ssdomg  8992  mbfid  25134  mthmpps  34511  brid  37113  extid  37117  cosscnvid  37289  idsymrel  37369
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