ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-er Structured version   GIF version

Definition df-er 6035
Description: Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6036 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6055, ersymb 6049, and ertr 6050. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
Assertion
Ref Expression
df-er (𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))

Detailed syntax breakdown of Definition df-er
StepHypRef Expression
1 cA . . 3 class A
2 cR . . 3 class 𝑅
31, 2wer 6032 . 2 wff 𝑅 Er A
42wrel 4292 . . 3 wff Rel 𝑅
52cdm 4287 . . . 4 class dom 𝑅
65, 1wceq 1242 . . 3 wff dom 𝑅 = A
72ccnv 4286 . . . . 5 class 𝑅
82, 2ccom 4291 . . . . 5 class (𝑅𝑅)
97, 8cun 2909 . . . 4 class (𝑅 ∪ (𝑅𝑅))
109, 2wss 2911 . . 3 wff (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅
114, 6, 10w3a 884 . 2 wff (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
123, 11wb 98 1 wff (𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
Colors of variables: wff set class
This definition is referenced by:  dfer2  6036  ereq1  6042  ereq2  6043  errel  6044  erdm  6045  ersym  6047  ertr  6050  xpiderm  6106
  Copyright terms: Public domain W3C validator