Theorem errel 6299

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 6290	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 958	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1289   ∪ cun 2997   ⊆ wss 2999  ^◡ccnv 4437  dom cdm 4438   ∘ ccom 4442  Rel wrel 4443   Er wer 6287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-3an 926  df-er 6290

This theorem is referenced by:  ercl  6301  ersym  6302  ertr  6305  ercnv  6311  erssxp  6313  erth  6334  iinerm  6362