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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 7433 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 7592. (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 7431 | . 2 class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
6 | vw | . . . . . . . . 9 setvar 𝑤 | |
7 | 6 | cv 1535 | . . . . . . . 8 class 𝑤 |
8 | 2 | cv 1535 | . . . . . . . . . 10 class 𝑥 |
9 | 3 | cv 1535 | . . . . . . . . . 10 class 𝑦 |
10 | 8, 9 | cop 4636 | . . . . . . . . 9 class 〈𝑥, 𝑦〉 |
11 | 4 | cv 1535 | . . . . . . . . 9 class 𝑧 |
12 | 10, 11 | cop 4636 | . . . . . . . 8 class 〈〈𝑥, 𝑦〉, 𝑧〉 |
13 | 7, 12 | wceq 1536 | . . . . . . 7 wff 𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 |
14 | 13, 1 | wa 395 | . . . . . 6 wff (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
15 | 14, 4 | wex 1775 | . . . . 5 wff ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
16 | 15, 3 | wex 1775 | . . . 4 wff ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
17 | 16, 2 | wex 1775 | . . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
18 | 17, 6 | cab 2711 | . 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
19 | 5, 18 | wceq 1536 | 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 7461 oprabid 7462 dfoprab2 7490 nfoprab1 7493 nfoprab2 7494 nfoprab3 7495 nfoprab 7496 oprabbid 7497 oprabbidv 7498 ssoprab2 7500 0mpo0 7515 mpo0 7517 cbvoprab2 7520 cbvoprab12v 7522 cbvoprab3v 7524 eloprabga 7540 eloprabgaOLD 7541 oprabrexex2 8001 eloprabi 8086 dftpos3 8267 join0 18462 meet0 18463 cnvoprabOLD 32737 mppspstlem 35555 mppsval 35556 colinearex 36041 cbvoprab1vw 36219 cbvoprab2vw 36220 cbvoprab123vw 36221 cbvoprab23vw 36222 cbvoprab13vw 36223 cbvoprab1davw 36253 cbvoprab2davw 36254 cbvoprab3davw 36255 cbvoprab123davw 36256 cbvoprab12davw 36257 cbvoprab23davw 36258 cbvoprab13davw 36259 csboprabg 37312 |
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