Theorem cnvi 6045

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.

Step	Hyp	Ref	Expression
1		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
2	1	ideq 5761	. . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
3		equcom 2021	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	2, 3	bitri 274	. . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦)
5	4	opabbii 5141	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6		df-cnv 5597	. 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7		df-id 5489	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
8	5, 6, 7	3eqtr4i 2776	1 ⊢ ◡ I = I

Syntax hints:   = wceq 1539   class class class wbr 5074  {copab 5136   I cid 5488  ^◡ccnv 5588

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597

This theorem is referenced by:  coi2  6167  funi  6466  cnvresid  6513  fcoi1  6648  ssdomg  8786  mbfid  24799  mthmpps  33544  brid  36442  extid  36446  cosscnvid  36599  idsymrel  36675