Theorem errel 8155

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 8146	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1138	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1525   ∪ cun 3863   ⊆ wss 3865  ^◡ccnv 5449  dom cdm 5450   ∘ ccom 5454  Rel wrel 5455   Er wer 8143

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 208  df-an 397  df-3an 1082  df-er 8146

This theorem is referenced by:  ercl  8157  ersym  8158  ertr  8161  ercnv  8167  erssxp  8169  erth  8195  iiner  8226  frgpuplem  18629  ismntop  30880  topfneec  33314  prter3  35570