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Definition df-oprab 6619
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 6618 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 6761. (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 6616 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6 vw . . . . . . . . 9 setvar 𝑤
76cv 1479 . . . . . . . 8 class 𝑤
82cv 1479 . . . . . . . . . 10 class 𝑥
93cv 1479 . . . . . . . . . 10 class 𝑦
108, 9cop 4161 . . . . . . . . 9 class 𝑥, 𝑦
114cv 1479 . . . . . . . . 9 class 𝑧
1210, 11cop 4161 . . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧
137, 12wceq 1480 . . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧
1413, 1wa 384 . . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1514, 4wex 1701 . . . . 5 wff 𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1615, 3wex 1701 . . . 4 wff 𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1716, 2wex 1701 . . 3 wff 𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1817, 6cab 2607 . 2 class {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
195, 18wceq 1480 1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Colors of variables: wff setvar class
This definition is referenced by:  oprabid  6642  dfoprab2  6666  nfoprab1  6669  nfoprab2  6670  nfoprab3  6671  nfoprab  6672  oprabbid  6673  ssoprab2  6676  mpt20  6690  cbvoprab2  6693  eloprabga  6712  oprabrexex2  7118  eloprabi  7192  dftpos3  7330  meet0  17077  join0  17078  cnvoprab  29382  mppspstlem  31229  mppsval  31230  colinearex  31862  csboprabg  32847
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