Theorem errel 8641

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 8632	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1145	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3903   ⊆ wss 3905  ^◡ccnv 5622  dom cdm 5623   ∘ ccom 5627  Rel wrel 5628   Er wer 8629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-er 8632

This theorem is referenced by:  ercl  8643  ersym  8644  ertr  8647  ercnv  8653  erssxp  8655  erth  8686  iiner  8723  eqg0el  19080  frgpuplem  19669  qusxpid  33313  ismntop  33995  topfneec  36331  prter3  38863