Theorem ercnv 8110

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 8098	. 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relcnv 5807	. . 3 ⊢ Rel ◡𝑅
3		id 22	. . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴)
4	3	ersymb 8103	. . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦))
5		vex 3418	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 3418	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 5603	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		df-br 4930	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 269	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
10		df-br 4930	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
11	4, 9, 10	3bitr3g 305	. . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
12	11	eqrelrdv2 5518	. . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
13	2, 12	mpanl1 687	. 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
14	1, 13	mpancom 675	1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2050  ⟨cop 4447   class class class wbr 4929  ^◡ccnv 5406  Rel wrel 5412   Er wer 8086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-er 8089

This theorem is referenced by:  errn  8111  prjspeclsp  38675