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

Step	Hyp	Ref	Expression
1		df-er 6172	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 954	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

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