Theorem errel 6431

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 6422	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 996	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1331   ∪ cun 3064   ⊆ wss 3066  ^◡ccnv 4533  dom cdm 4534   ∘ ccom 4538  Rel wrel 4539   Er wer 6419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-3an 964  df-er 6422

This theorem is referenced by:  ercl  6433  ersym  6434  ertr  6437  ercnv  6443  erssxp  6445  erth  6466  iinerm  6494