Theorem ercnv 8725

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 8713	. 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relcnv 6111	. . 3 ⊢ Rel ◡𝑅
3		id 23	. . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴)
4	3	ersymb 8718	. . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦))
5		vex 3462	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 3462	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 5873	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		df-br 5115	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 280	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
10		df-br 5115	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
11	4, 9, 10	3bitr3g 316	. . . 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 713	. 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
14	1, 13	mpancom 701	1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   class class class wbr 5114  ^◡ccnv 5665  Rel wrel 5671   Er wer 8700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-er 8703

This theorem is used by:  errn  8726  prjspeclsp  43385