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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 4064 | . 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 3596 | . . . . . . 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 4068 opabbid 4069 nfopab 4072 nfopab1 4073 nfopab2 4074 cbvopab 4075 cbvopab1 4077 cbvopab2 4078 cbvopab1s 4079 cbvopab2v 4081 unopab 4083 opabid 4258 elopab 4259 ssopab2 4276 iunopab 4282 elxpi 4643 rabxp 4664 csbxpg 4708 relopabi 4753 opabbrex 5919 dfoprab2 5922 dmoprab 5956 dfopab2 6190 cnvoprab 6235 |
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