Theorem errel 6596

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 6587	. 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 3151   ⊆ wss 3153  ^◡ccnv 4658  dom cdm 4659   ∘ ccom 4663  Rel wrel 4664   Er wer 6584

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 6587

This theorem is referenced by:  ercl  6598  ersym  6599  ertr  6602  ercnv  6608  erssxp  6610  erth  6633  iinerm  6661  eqg0el  13299