Theorem errel 6138

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 6129	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 953	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1284   ∪ cun 2971   ⊆ wss 2973  ^◡ccnv 4362  dom cdm 4363   ∘ ccom 4367  Rel wrel 4368   Er wer 6126

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-3an 921  df-er 6129

This theorem is referenced by:  ercl  6140  ersym  6141  ertr  6144  ercnv  6150  erssxp  6152  erth  6173  iinerm  6201