Theorem errel 8313

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 8304	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1142	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1538   ∪ cun 3858   ⊆ wss 3860  ^◡ccnv 5526  dom cdm 5527   ∘ ccom 5531  Rel wrel 5532   Er wer 8301

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 210  df-an 400  df-3an 1086  df-er 8304

This theorem is referenced by:  ercl  8315  ersym  8316  ertr  8319  ercnv  8325  erssxp  8327  erth  8353  iiner  8384  frgpuplem  18970  eqg0el  31082  qusxpid  31084  ismntop  31499  topfneec  34119  prter3  36484