Theorem cnvi 6003

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 3500	. . . . 5 ⊢ 𝑥 ∈ V
2	1	ideq 5726	. . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
3		equcom 2024	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	2, 3	bitri 277	. . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦)
5	4	opabbii 5136	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
6		df-cnv 5566	. 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥}
7		df-id 5463	. 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
8	5, 6, 7	3eqtr4i 2857	1 ⊢ ◡ I = I

Syntax hints:   = wceq 1536   class class class wbr 5069  {copab 5131   I cid 5462  ^◡ccnv 5557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566

This theorem is referenced by:  coi2  6119  funi  6390  cnvresid  6436  fcoi1  6555  ssdomg  8558  mbfid  24239  mthmpps  32833  brid  35568  extid  35572  cosscnvid  35725  idsymrel  35801