Theorem errel 8465

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 8456	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1143	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3881   ⊆ wss 3883  ^◡ccnv 5579  dom cdm 5580   ∘ ccom 5584  Rel wrel 5585   Er wer 8453

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 206  df-an 396  df-3an 1087  df-er 8456

This theorem is referenced by:  ercl  8467  ersym  8468  ertr  8471  ercnv  8477  erssxp  8479  erth  8505  iiner  8536  frgpuplem  19293  eqg0el  31459  qusxpid  31461  ismntop  31876  topfneec  34471  prter3  36823