Theorem errel 8706

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 8696	. 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   ∪ cun 3897   ⊆ wss 3899  ^◡ccnv 5654  dom cdm 5655   ∘ ccom 5659  Rel wrel 5660   Er wer 8693

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 8696

This theorem is used by:  ercl  8708  ersym  8709  ertr  8712  ercnv  8718  erssxp  8720  erth  8751  iiner  8789  qusxpid  19308  eqg0el  19311  frgpuplem  19899  ismntop  34536  topfneec  36974  prter3  39755