Theorem errel 8680

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 8671	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1145	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3912   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638   ∘ ccom 5642  Rel wrel 5643   Er wer 8668

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 207  df-an 396  df-3an 1088  df-er 8671

This theorem is referenced by:  ercl  8682  ersym  8683  ertr  8686  ercnv  8692  erssxp  8694  erth  8725  iiner  8762  eqg0el  19115  frgpuplem  19702  qusxpid  33334  ismntop  34016  topfneec  36343  prter3  38875