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Theorem errel 8155
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 8146 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 1138 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1525  cun 3863  wss 3865  ccnv 5449  dom cdm 5450  ccom 5454  Rel wrel 5455   Er wer 8143
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 208  df-an 397  df-3an 1082  df-er 8146
This theorem is referenced by:  ercl  8157  ersym  8158  ertr  8161  ercnv  8167  erssxp  8169  erth  8195  iiner  8226  frgpuplem  18629  ismntop  30880  topfneec  33314  prter3  35570
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