Theorem errel 8718

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 8709	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1144	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3946   ⊆ wss 3948  ^◡ccnv 5675  dom cdm 5676   ∘ ccom 5680  Rel wrel 5681   Er wer 8706

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 206  df-an 396  df-3an 1088  df-er 8709

This theorem is referenced by:  ercl  8720  ersym  8721  ertr  8724  ercnv  8730  erssxp  8732  erth  8758  iiner  8789  frgpuplem  19685  eqg0el  32762  qusxpid  32764  ismntop  33319  topfneec  35556  prter3  38068