Theorem ercnv 6613

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 6601	. 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)
2		relcnv 5047	. . 3 ⊢ Rel ◡𝑅
3		id 19	. . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴)
4	3	ersymb 6606	. . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦))
5		vex 2766	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 2766	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brcnv 4849	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8		df-br 4034	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
9	7, 8	bitr3i 186	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
10		df-br 4034	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
11	4, 9, 10	3bitr3g 222	. . . 4 ⊢ (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
12	11	eqrelrdv2 4762	. . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
13	2, 12	mpanl1 434	. 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅)
14	1, 13	mpancom 422	1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  ⟨cop 3625   class class class wbr 4033  ^◡ccnv 4662  Rel wrel 4668   Er wer 6589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-er 6592

This theorem is referenced by:  errn  6614