Theorem errel 7638

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 7629	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1069	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538   ⊆ wss 3540  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043   Er wer 7626

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 196  df-an 385  df-3an 1033  df-er 7629

This theorem is referenced by:  ercl  7640  ersym  7641  ertr  7644  ercnv  7650  erssxp  7652  erth  7678  iiner  7706  frgpuplem  18008  ismntop  29398  topfneec  31520  prter3  33185