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

Step	Hyp	Ref	Expression
1		df-er 8746	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
2	1	simp1bi 1145	1 ⊢ (𝑅 Er 𝐴 → Rel 𝑅)

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