Theorem errel 6706

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 6697	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1036	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1395   ∪ cun 3196   ⊆ wss 3198  ^◡ccnv 4722  dom cdm 4723   ∘ ccom 4727  Rel wrel 4728   Er wer 6694

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 1004  df-er 6697

This theorem is referenced by:  ercl  6708  ersym  6709  ertr  6712  ercnv  6718  erssxp  6720  erth  6743  iinerm  6771  eqg0el  13809