Theorem errel 8700

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 8690	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1161	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1567   ∪ cun 3911   ⊆ wss 3913  ^◡ccnv 5658  dom cdm 5659   ∘ ccom 5663  Rel wrel 5664   Er wer 8687

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 210  df-an 401  df-3an 1103  df-er 8690

This theorem is referenced by:  ercl  8702  ersym  8703  ertr  8706  ercnv  8712  erssxp  8714  erth  8745  iiner  8783  qusxpid  19247  eqg0el  19250  frgpuplem  19838  ismntop  34357  topfneec  36751  prter3  39541