Theorem errel 8733

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 8724	. 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 3929   ⊆ wss 3931  ^◡ccnv 5658  dom cdm 5659   ∘ ccom 5663  Rel wrel 5664   Er wer 8721

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 8724

This theorem is referenced by:  ercl  8735  ersym  8736  ertr  8739  ercnv  8745  erssxp  8747  erth  8775  iiner  8808  eqg0el  19171  frgpuplem  19758  qusxpid  33383  ismntop  34062  topfneec  36378  prter3  38905