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Definition df-oprab 7288
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 7287 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 7442. (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 7285 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6 vw . . . . . . . . 9 setvar 𝑤
76cv 1538 . . . . . . . 8 class 𝑤
82cv 1538 . . . . . . . . . 10 class 𝑥
93cv 1538 . . . . . . . . . 10 class 𝑦
108, 9cop 4568 . . . . . . . . 9 class 𝑥, 𝑦
114cv 1538 . . . . . . . . 9 class 𝑧
1210, 11cop 4568 . . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧
137, 12wceq 1539 . . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧
1413, 1wa 396 . . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1514, 4wex 1782 . . . . 5 wff 𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1615, 3wex 1782 . . . 4 wff 𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1716, 2wex 1782 . . 3 wff 𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1817, 6cab 2716 . 2 class {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
195, 18wceq 1539 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  7315  oprabid  7316  dfoprab2  7342  nfoprab1  7345  nfoprab2  7346  nfoprab3  7347  nfoprab  7348  oprabbid  7349  oprabbidv  7350  ssoprab2  7352  0mpo0  7367  mpo0  7369  cbvoprab2  7372  eloprabga  7391  eloprabgaOLD  7392  oprabrexex2  7830  eloprabi  7912  dftpos3  8069  join0  18132  meet0  18133  cnvoprabOLD  31064  mppspstlem  33542  mppsval  33543  colinearex  34371  csboprabg  35510
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