Theorem errel 8682

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 8672	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1157	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1559   ∪ cun 3900   ⊆ wss 3902  ^◡ccnv 5642  dom cdm 5643   ∘ ccom 5647  Rel wrel 5648   Er wer 8669

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 209  df-an 400  df-3an 1099  df-er 8672

This theorem is referenced by:  ercl  8684  ersym  8685  ertr  8688  ercnv  8694  erssxp  8696  erth  8727  iiner  8765  eqg0el  19215  frgpuplem  19803  qusxpid  33510  ismntop  34284  topfneec  36676  prter3  39467