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Definition df-oprab 7149
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 7148 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 ovmpo 7299. (Contributed by NM, 12-Mar-1995.)
Assertion
Ref Expression
df-oprab {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Distinct variable groups:   𝑥,𝑤   𝑦,𝑤   𝑧,𝑤   𝜑,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Detailed syntax breakdown of Definition df-oprab
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 vz . . 3 setvar 𝑧
51, 2, 3, 4coprab 7146 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6 vw . . . . . . . . 9 setvar 𝑤
76cv 1527 . . . . . . . 8 class 𝑤
82cv 1527 . . . . . . . . . 10 class 𝑥
93cv 1527 . . . . . . . . . 10 class 𝑦
108, 9cop 4565 . . . . . . . . 9 class 𝑥, 𝑦
114cv 1527 . . . . . . . . 9 class 𝑧
1210, 11cop 4565 . . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧
137, 12wceq 1528 . . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧
1413, 1wa 396 . . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1514, 4wex 1771 . . . . 5 wff 𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1615, 3wex 1771 . . . 4 wff 𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1716, 2wex 1771 . . 3 wff 𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1817, 6cab 2799 . 2 class {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
195, 18wceq 1528 1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Colors of variables: wff setvar class
This definition is referenced by:  oprabidw  7176  oprabid  7177  dfoprab2  7201  nfoprab1  7204  nfoprab2  7205  nfoprab3  7206  nfoprab  7207  oprabbid  7208  ssoprab2  7211  0mpo0  7226  mpo0  7228  cbvoprab2  7231  eloprabga  7250  oprabrexex2  7670  eloprabi  7752  dftpos3  7901  meet0  17737  join0  17738  cnvoprabOLD  30383  mppspstlem  32716  mppsval  32717  colinearex  33419  csboprabg  34494
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