Theorem cnvi 6100

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 3434	. . . . 5 ⊢ 𝑥 ∈ V
2	1	ideq 5802	. . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
3		equcom 2020	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	2, 3	bitri 275	. . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦)
5	4	opabbii 5153	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6		df-cnv 5633	. 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7		df-id 5520	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
8	5, 6, 7	3eqtr4i 2770	1 ⊢ ◡ I = I

Syntax hints:   = wceq 1542   class class class wbr 5086  {copab 5148   I cid 5519  ^◡ccnv 5624

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633

This theorem is referenced by:  coi2  6223  funi  6525  cnvresid  6572  fcoi1  6709  f1oi  6813  ssdomg  8941  mbfid  25615  mthmpps  35783  brid  38650  extid  38654  cosscnvid  38909  idsymrel  38983