Theorem ercnv 8721

Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
ercnv	⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)

Proof of Theorem ercnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		errel 8709	. 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relcnv 6094	. . 3 ⊢ Rel ◡𝑅
3		id 22	. . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴)
4	3	ersymb 8714	. . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦))
5		vex 3470	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 3470	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 5873	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		df-br 5140	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 277	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
10		df-br 5140	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
11	4, 9, 10	3bitr3g 313	. . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
12	11	eqrelrdv2 5786	. . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
13	2, 12	mpanl1 697	. 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
14	1, 13	mpancom 685	1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ⟨cop 4627   class class class wbr 5139  ^◡ccnv 5666  Rel wrel 5672   Er wer 8697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-er 8700

This theorem is referenced by:  errn  8722  prjspeclsp  41904