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Definition df-er 6042
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 6043 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6062, ersymb 6056, and ertr 6057. (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 6039 . 2 wff 𝑅 Er A
42wrel 4293 . . 3 wff Rel 𝑅
52cdm 4288 . . . 4 class dom 𝑅
65, 1wceq 1242 . . 3 wff dom 𝑅 = A
72ccnv 4287 . . . . 5 class 𝑅
82, 2ccom 4292 . . . . 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  6043  ereq1  6049  ereq2  6050  errel  6051  erdm  6052  ersym  6054  ertr  6057  xpiderm  6113
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