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Theorem errel 7904
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 7895 . 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 1630  cun 3719  wss 3721  ccnv 5248  dom cdm 5249  ccom 5253  Rel wrel 5254   Er wer 7892
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 383  df-3an 1072  df-er 7895
This theorem is referenced by:  ercl  7906  ersym  7907  ertr  7910  ercnv  7916  erssxp  7918  erth  7942  iiner  7970  frgpuplem  18391  ismntop  30404  topfneec  32681  prter3  34683
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