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Theorem cnvi 6098
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 3451 . . . . 5 𝑥 ∈ V
21ideq 5812 . . . 4 (𝑦 I 𝑥𝑦 = 𝑥)
3 equcom 2022 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
42, 3bitri 275 . . 3 (𝑦 I 𝑥𝑥 = 𝑦)
54opabbii 5176 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6 df-cnv 5645 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7 df-id 5535 . 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 5109  {copab 5171   I cid 5534  ccnv 5636
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645
This theorem is referenced by:  coi2  6219  funi  6537  cnvresid  6584  fcoi1  6720  ssdomg  8946  mbfid  25022  mthmpps  34240  brid  36817  extid  36821  cosscnvid  36993  idsymrel  37073
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