Theorem errel 8765

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 8755	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1163	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∪ cun 4152  ^◡ccnv 5657  dom cdm 5658   ∘ ccom 5662  Rel wrel 5663   Er wer 8752

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-er 8755

This theorem is used by:  ercl  8767  ersym  8768  ertr  8771  ercnv  8777  erssxp  8779  erth  8810  iiner  8848  qusxpid  19445  eqg0el  19448  frgpuplem  20037  ismntop  34769  topfneec  37358  prter3  40205