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Theorem errel 8755
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 8746 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 1145 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cun 3948  wss 3950  ccnv 5683  dom cdm 5684  ccom 5688  Rel wrel 5689   Er wer 8743
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 207  df-an 396  df-3an 1088  df-er 8746
This theorem is referenced by:  ercl  8757  ersym  8758  ertr  8761  ercnv  8767  erssxp  8769  erth  8797  iiner  8830  eqg0el  19202  frgpuplem  19791  qusxpid  33392  ismntop  34028  topfneec  36357  prter3  38884
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