Theorem errel 8301

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 8292	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1141	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1536   ∪ cun 3937   ⊆ wss 3939  ^◡ccnv 5557  dom cdm 5558   ∘ ccom 5562  Rel wrel 5563   Er wer 8289

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 209  df-an 399  df-3an 1085  df-er 8292

This theorem is referenced by:  ercl  8303  ersym  8304  ertr  8307  ercnv  8313  erssxp  8315  erth  8341  iiner  8372  frgpuplem  18901  eqg0el  30930  qusxpid  30932  ismntop  31271  topfneec  33707  prter3  36022