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

Step	Hyp	Ref	Expression
1		df-er 7895	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1138	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

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