Theorem errel 8643

Description: An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
errel	⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Proof of Theorem errel

Step	Hyp	Ref	Expression
1		df-er 8633	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1151	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1547   ∪ cun 3881   ⊆ wss 3883  ^◡ccnv 5617  dom cdm 5618   ∘ ccom 5622  Rel wrel 5623   Er wer 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-er 8633

This theorem is referenced by:  ercl  8645  ersym  8646  ertr  8649  ercnv  8655  erssxp  8657  erth  8688  iiner  8726  eqg0el  19149  frgpuplem  19738  qusxpid  33446  ismntop  34210  topfneec  36583  prter3  39374