Theorem errel 8507

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 8498	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1144	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3885   ⊆ wss 3887  ^◡ccnv 5588  dom cdm 5589   ∘ ccom 5593  Rel wrel 5594   Er wer 8495

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 206  df-an 397  df-3an 1088  df-er 8498

This theorem is referenced by:  ercl  8509  ersym  8510  ertr  8513  ercnv  8519  erssxp  8521  erth  8547  iiner  8578  frgpuplem  19378  eqg0el  31557  qusxpid  31559  ismntop  31976  topfneec  34544  prter3  36896