Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  errel Structured version   Visualization version   GIF version

Theorem errel 8313
 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 8304 . 2 (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp1bi 1142 1 (𝑅 Er 𝐴 → Rel 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∪ cun 3858   ⊆ wss 3860  ◡ccnv 5526  dom cdm 5527   ∘ ccom 5531  Rel wrel 5532   Er wer 8301 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 210  df-an 400  df-3an 1086  df-er 8304 This theorem is referenced by:  ercl  8315  ersym  8316  ertr  8319  ercnv  8325  erssxp  8327  erth  8353  iiner  8384  frgpuplem  18970  eqg0el  31082  qusxpid  31084  ismntop  31499  topfneec  34119  prter3  36484
 Copyright terms: Public domain W3C validator