Theorem errel 8644

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 8635	. 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 3899   ⊆ wss 3901  ^◡ccnv 5623  dom cdm 5624   ∘ ccom 5628  Rel wrel 5629   Er wer 8632

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 8635

This theorem is referenced by:  ercl  8646  ersym  8647  ertr  8650  ercnv  8656  erssxp  8658  erth  8689  iiner  8726  eqg0el  19112  frgpuplem  19701  qusxpid  33444  ismntop  34183  topfneec  36549  prter3  39142