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Mirrors > Home > ILE Home > Th. List > df-opab | GIF version |
Description: Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
df-opab | ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | 1, 2, 3 | copab 4063 | . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
5 | vz | . . . . . . . 8 setvar 𝑧 | |
6 | 5 | cv 1352 | . . . . . . 7 class 𝑧 |
7 | 2 | cv 1352 | . . . . . . . 8 class 𝑥 |
8 | 3 | cv 1352 | . . . . . . . 8 class 𝑦 |
9 | 7, 8 | cop 3595 | . . . . . . 7 class ⟨𝑥, 𝑦⟩ |
10 | 6, 9 | wceq 1353 | . . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦⟩ |
11 | 10, 1 | wa 104 | . . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
12 | 11, 3 | wex 1492 | . . . 4 wff ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
13 | 12, 2 | wex 1492 | . . 3 wff ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
14 | 13, 5 | cab 2163 | . 2 class {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} |
15 | 4, 14 | wceq 1353 | 1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} |
Colors of variables: wff set class |
This definition is referenced by: opabss 4067 opabbid 4068 nfopab 4071 nfopab1 4072 nfopab2 4073 cbvopab 4074 cbvopab1 4076 cbvopab2 4077 cbvopab1s 4078 cbvopab2v 4080 unopab 4082 opabid 4257 elopab 4258 ssopab2 4275 iunopab 4281 elxpi 4642 rabxp 4663 csbxpg 4707 relopabi 4752 opabbrex 5918 dfoprab2 5921 dmoprab 5955 dfopab2 6189 cnvoprab 6234 |
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