Theorem errel 7751

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 7742	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1076	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1483   ∪ cun 3572   ⊆ wss 3574  ^◡ccnv 5113  dom cdm 5114   ∘ ccom 5118  Rel wrel 5119   Er wer 7739

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 197  df-an 386  df-3an 1039  df-er 7742

This theorem is referenced by:  ercl  7753  ersym  7754  ertr  7757  ercnv  7763  erssxp  7765  erth  7791  iiner  7819  frgpuplem  18185  ismntop  30070  topfneec  32350  prter3  34167