Theorem errel 6558

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 6549	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1013	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1363   ∪ cun 3139   ⊆ wss 3141  ^◡ccnv 4637  dom cdm 4638   ∘ ccom 4642  Rel wrel 4643   Er wer 6546

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 981  df-er 6549

This theorem is referenced by:  ercl  6560  ersym  6561  ertr  6564  ercnv  6570  erssxp  6572  erth  6593  iinerm  6621