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Mirrors > Home > ILE Home > Th. List > df-er | GIF version |
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 6130 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6149, ersymb 6143, and ertr 6144. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.) |
Ref | Expression |
---|---|
df-er | ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | wer 6126 | . 2 wff 𝑅 Er 𝐴 |
4 | 2 | wrel 4368 | . . 3 wff Rel 𝑅 |
5 | 2 | cdm 4363 | . . . 4 class dom 𝑅 |
6 | 5, 1 | wceq 1284 | . . 3 wff dom 𝑅 = 𝐴 |
7 | 2 | ccnv 4362 | . . . . 5 class ◡𝑅 |
8 | 2, 2 | ccom 4367 | . . . . 5 class (𝑅 ∘ 𝑅) |
9 | 7, 8 | cun 2971 | . . . 4 class (◡𝑅 ∪ (𝑅 ∘ 𝑅)) |
10 | 9, 2 | wss 2973 | . . 3 wff (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 |
11 | 4, 6, 10 | w3a 919 | . 2 wff (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) |
12 | 3, 11 | wb 103 | 1 wff (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) |
Colors of variables: wff set class |
This definition is referenced by: dfer2 6130 ereq1 6136 ereq2 6137 errel 6138 erdm 6139 ersym 6141 ertr 6144 xpiderm 6200 |
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