Theorem errel 8639

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 8630	. 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 1541   ∪ cun 3896   ⊆ wss 3898  ^◡ccnv 5620  dom cdm 5621   ∘ ccom 5625  Rel wrel 5626   Er wer 8627

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 8630

This theorem is referenced by:  ercl  8641  ersym  8642  ertr  8645  ercnv  8651  erssxp  8653  erth  8684  iiner  8721  eqg0el  19099  frgpuplem  19688  qusxpid  33337  ismntop  34062  topfneec  36422  prter3  39004