Theorem errel 8753

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 8744	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1144	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1537   ∪ cun 3961   ⊆ wss 3963  ^◡ccnv 5688  dom cdm 5689   ∘ ccom 5693  Rel wrel 5694   Er wer 8741

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 8744

This theorem is referenced by:  ercl  8755  ersym  8756  ertr  8759  ercnv  8765  erssxp  8767  erth  8795  iiner  8828  eqg0el  19214  frgpuplem  19805  qusxpid  33371  ismntop  33989  topfneec  36338  prter3  38864