Theorem errel 8655

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 8645	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1146	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3901   ⊆ wss 3903  ^◡ccnv 5631  dom cdm 5632   ∘ ccom 5636  Rel wrel 5637   Er wer 8642

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 8645

This theorem is referenced by:  ercl  8657  ersym  8658  ertr  8661  ercnv  8667  erssxp  8669  erth  8700  iiner  8738  eqg0el  19124  frgpuplem  19713  qusxpid  33455  ismntop  34203  topfneec  36568  prter3  39255