Theorem errel 8702

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 8692	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1162	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∪ cun 3902   ⊆ wss 3904  ^◡ccnv 5659  dom cdm 5660   ∘ ccom 5664  Rel wrel 5665   Er wer 8689

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-er 8692

This theorem is used by:  ercl  8704  ersym  8705  ertr  8708  ercnv  8714  erssxp  8716  erth  8747  iiner  8785  qusxpid  19257  eqg0el  19260  frgpuplem  19848  ismntop  34425  topfneec  36894  prter3  39684