Theorem errel 6598

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 6589	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1014	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3152   ⊆ wss 3154  ^◡ccnv 4659  dom cdm 4660   ∘ ccom 4664  Rel wrel 4665   Er wer 6586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-3an 982  df-er 6589

This theorem is referenced by:  ercl  6600  ersym  6601  ertr  6604  ercnv  6610  erssxp  6612  erth  6635  iinerm  6663  eqg0el  13302