Theorem errel 6810

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 6801	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1043	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1402   ∪ cun 3218   ⊆ wss 3220  ^◡ccnv 4771  dom cdm 4772   ∘ ccom 4776  Rel wrel 4777   Er wer 6798

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 1011  df-er 6801

This theorem is referenced by:  ercl  6812  ersym  6813  ertr  6816  ercnv  6822  erssxp  6824  erth  6847  iinerm  6875  eqg0el  14015