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Theorem errel 7711
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 7702 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 1074 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cun 3558  wss 3560  ccnv 5083  dom cdm 5084  ccom 5088  Rel wrel 5089   Er wer 7699
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 197  df-an 386  df-3an 1038  df-er 7702
This theorem is referenced by:  ercl  7713  ersym  7714  ertr  7717  ercnv  7723  erssxp  7725  erth  7751  iiner  7779  frgpuplem  18125  ismntop  29894  topfneec  32045  prter3  33686
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