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Theorem errel 6181
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 6172 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 954 1 (𝑅 Er 𝐴 → Rel 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cun 2972  wss 2974  ccnv 4370  dom cdm 4371  ccom 4375  Rel wrel 4376   Er wer 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104
This theorem depends on definitions:  df-bi 115  df-3an 922  df-er 6172
This theorem is referenced by:  ercl  6183  ersym  6184  ertr  6187  ercnv  6193  erssxp  6195  erth  6216  iinerm  6244
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