Theorem errel 8734

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 8724	. 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 1569   ⊆ wss 3896   ∪ cun 4148  ^◡ccnv 5650  dom cdm 5651   ∘ ccom 5655  Rel wrel 5656   Er wer 8721

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 8724

This theorem is used by:  ercl  8736  ersym  8737  ertr  8740  ercnv  8746  erssxp  8748  erth  8779  iiner  8817  qusxpid  19404  eqg0el  19407  frgpuplem  19995  ismntop  34708  topfneec  37239  prter3  40048