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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 3478 . . . . 5 𝑥 ∈ V
21ideq 5852 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2021 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 274 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5684 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5574 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2770 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   class class class wbr 5148  {copab 5210   I cid 5573  ccnv 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684
This theorem is referenced by:  coi2  6262  funi  6580  cnvresid  6627  fcoi1  6765  ssdomg  8995  mbfid  25151  mthmpps  34568  brid  37170  extid  37174  cosscnvid  37346  idsymrel  37426
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