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Theorem errel 6406
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
StepHypRef Expression
1 df-er 6397 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 981 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  cun 3039  wss 3041  ccnv 4508  dom cdm 4509  ccom 4513  Rel wrel 4514   Er wer 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105
This theorem depends on definitions:  df-bi 116  df-3an 949  df-er 6397
This theorem is referenced by:  ercl  6408  ersym  6409  ertr  6412  ercnv  6418  erssxp  6420  erth  6441  iinerm  6469
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