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Theorem cnvi 5999
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 3497 . . . . 5 𝑥 ∈ V
21ideq 5722 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2021 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 277 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5132 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5562 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5459 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
85, 6, 73eqtr4i 2854 1 I = I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   class class class wbr 5065  {copab 5127   I cid 5458  ccnv 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562
This theorem is referenced by:  coi2  6115  funi  6386  cnvresid  6432  fcoi1  6551  ssdomg  8554  mbfid  24235  mthmpps  32829  brid  35563  extid  35567  cosscnvid  35720  idsymrel  35796
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