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Theorem errel 8287
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 8278 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 1137 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cun 3931  wss 3933  ccnv 5547  dom cdm 5548  ccom 5552  Rel wrel 5553   Er wer 8275
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 1081  df-er 8278
This theorem is referenced by:  ercl  8289  ersym  8290  ertr  8293  ercnv  8299  erssxp  8301  erth  8327  iiner  8358  frgpuplem  18827  eqg0el  30853  qusxpid  30855  ismntop  31166  topfneec  33600  prter3  35898
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