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Mirrors > Home > MPE Home > Th. List > df-oprab | Structured version Visualization version GIF version |
Description: Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally 𝑥, 𝑦, and 𝑧 are distinct, although the definition doesn't strictly require it. See df-ov 7216 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of an operation given by a class abstraction is given by ovmpo 7369. (Contributed by NM, 12-Mar-1995.) |
Ref | Expression |
---|---|
df-oprab | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | vz | . . 3 setvar 𝑧 | |
5 | 1, 2, 3, 4 | coprab 7214 | . 2 class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
6 | vw | . . . . . . . . 9 setvar 𝑤 | |
7 | 6 | cv 1542 | . . . . . . . 8 class 𝑤 |
8 | 2 | cv 1542 | . . . . . . . . . 10 class 𝑥 |
9 | 3 | cv 1542 | . . . . . . . . . 10 class 𝑦 |
10 | 8, 9 | cop 4547 | . . . . . . . . 9 class 〈𝑥, 𝑦〉 |
11 | 4 | cv 1542 | . . . . . . . . 9 class 𝑧 |
12 | 10, 11 | cop 4547 | . . . . . . . 8 class 〈〈𝑥, 𝑦〉, 𝑧〉 |
13 | 7, 12 | wceq 1543 | . . . . . . 7 wff 𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 |
14 | 13, 1 | wa 399 | . . . . . 6 wff (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
15 | 14, 4 | wex 1787 | . . . . 5 wff ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
16 | 15, 3 | wex 1787 | . . . 4 wff ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
17 | 16, 2 | wex 1787 | . . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
18 | 17, 6 | cab 2714 | . 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
19 | 5, 18 | wceq 1543 | 1 wff {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
Colors of variables: wff setvar class |
This definition is referenced by: oprabidw 7244 oprabid 7245 dfoprab2 7269 nfoprab1 7272 nfoprab2 7273 nfoprab3 7274 nfoprab 7275 oprabbid 7276 ssoprab2 7279 0mpo0 7294 mpo0 7296 cbvoprab2 7299 eloprabga 7318 eloprabgaOLD 7319 oprabrexex2 7751 eloprabi 7833 dftpos3 7986 join0 17911 meet0 17912 cnvoprabOLD 30775 mppspstlem 33246 mppsval 33247 colinearex 34099 csboprabg 35238 |
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