Theorem ercnv 8519

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 8507	. 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relcnv 6012	. . 3 ⊢ Rel ◡𝑅
3		id 22	. . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴)
4	3	ersymb 8512	. . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦))
5		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 3436	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 5791	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		df-br 5075	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 276	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
10		df-br 5075	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
11	4, 9, 10	3bitr3g 313	. . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
12	11	eqrelrdv2 5705	. . 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 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  ^◡ccnv 5588  Rel wrel 5594   Er wer 8495

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-xp 5595  df-rel 5596  df-cnv 5597  df-er 8498

This theorem is referenced by:  errn  8520  prjspeclsp  40451