Theorem errel 6544

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 6535	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1012	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1353   ∪ cun 3128   ⊆ wss 3130  ^◡ccnv 4626  dom cdm 4627   ∘ ccom 4631  Rel wrel 4632   Er wer 6532

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 980  df-er 6535

This theorem is referenced by:  ercl  6546  ersym  6547  ertr  6550  ercnv  6556  erssxp  6558  erth  6579  iinerm  6607