Theorem errel 6641

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 6632	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1015	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1373   ∪ cun 3168   ⊆ wss 3170  ^◡ccnv 4681  dom cdm 4682   ∘ ccom 4686  Rel wrel 4687   Er wer 6629

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 983  df-er 6632

This theorem is referenced by:  ercl  6643  ersym  6644  ertr  6647  ercnv  6653  erssxp  6655  erth  6678  iinerm  6706  eqg0el  13635