Theorem errel 8772

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 8763	. 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 1537   ∪ cun 3974   ⊆ wss 3976  ^◡ccnv 5699  dom cdm 5700   ∘ ccom 5704  Rel wrel 5705   Er wer 8760

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 1089  df-er 8763

This theorem is referenced by:  ercl  8774  ersym  8775  ertr  8778  ercnv  8784  erssxp  8786  erth  8814  iiner  8847  eqg0el  19223  frgpuplem  19814  qusxpid  33356  ismntop  33972  topfneec  36321  prter3  38838