Theorem errel 8710

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 8700	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1163	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3904   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663   ∘ ccom 5667  Rel wrel 5668   Er wer 8697

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 402  df-3an 1105  df-er 8700

This theorem is used by:  ercl  8712  ersym  8713  ertr  8716  ercnv  8722  erssxp  8724  erth  8755  iiner  8793  qusxpid  19295  eqg0el  19298  frgpuplem  19886  ismntop  34480  topfneec  36923  prter3  39714