Theorem cnvi 4879

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 2644	. . . . 5 ⊢ 𝑥 ∈ V
2	1	ideq 4629	. . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
3		equcom 1650	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	2, 3	bitri 183	. . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦)
5	4	opabbii 3935	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6		df-cnv 4485	. 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7		df-id 4153	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
8	5, 6, 7	3eqtr4i 2130	1 ⊢ ◡ I = I

Syntax hints:   = wceq 1299   class class class wbr 3875  {copab 3928   I cid 4148  ^◡ccnv 4476

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485

This theorem is referenced by:  coi2  4991  funi  5091  cnvresid  5133  fcoi1  5239  ssdomg  6602