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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 6925 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 the most common operation class builder is given by ovmpt2 7073. (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 6923 | . 2 class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
6 | vw | . . . . . . . . 9 setvar 𝑤 | |
7 | 6 | cv 1600 | . . . . . . . 8 class 𝑤 |
8 | 2 | cv 1600 | . . . . . . . . . 10 class 𝑥 |
9 | 3 | cv 1600 | . . . . . . . . . 10 class 𝑦 |
10 | 8, 9 | cop 4404 | . . . . . . . . 9 class 〈𝑥, 𝑦〉 |
11 | 4 | cv 1600 | . . . . . . . . 9 class 𝑧 |
12 | 10, 11 | cop 4404 | . . . . . . . 8 class 〈〈𝑥, 𝑦〉, 𝑧〉 |
13 | 7, 12 | wceq 1601 | . . . . . . 7 wff 𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 |
14 | 13, 1 | wa 386 | . . . . . 6 wff (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
15 | 14, 4 | wex 1823 | . . . . 5 wff ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
16 | 15, 3 | wex 1823 | . . . 4 wff ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
17 | 16, 2 | wex 1823 | . . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
18 | 17, 6 | cab 2763 | . 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
19 | 5, 18 | wceq 1601 | 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: oprabid 6953 dfoprab2 6978 nfoprab1 6981 nfoprab2 6982 nfoprab3 6983 nfoprab 6984 oprabbid 6985 ssoprab2 6988 mpt20 7002 cbvoprab2 7005 eloprabga 7024 oprabrexex2 7435 eloprabi 7512 dftpos3 7652 meet0 17523 join0 17524 cnvoprabOLD 30064 mppspstlem 32067 mppsval 32068 colinearex 32756 csboprabg 33772 |
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