Theorem errel 8631

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 8622	. 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 1541   ∪ cun 3900   ⊆ wss 3902  ^◡ccnv 5615  dom cdm 5616   ∘ ccom 5620  Rel wrel 5621   Er wer 8619

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 8622

This theorem is referenced by:  ercl  8633  ersym  8634  ertr  8637  ercnv  8643  erssxp  8645  erth  8676  iiner  8713  eqg0el  19096  frgpuplem  19685  qusxpid  33326  ismntop  34037  topfneec  36395  prter3  38927