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Definition df-oprab 5494
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 5493 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 5614. (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 5491 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6 vw . . . . . . . . 9 setvar 𝑤
76cv 1242 . . . . . . . 8 class 𝑤
82cv 1242 . . . . . . . . . 10 class 𝑥
93cv 1242 . . . . . . . . . 10 class 𝑦
108, 9cop 3375 . . . . . . . . 9 class 𝑥, 𝑦
114cv 1242 . . . . . . . . 9 class 𝑧
1210, 11cop 3375 . . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧
137, 12wceq 1243 . . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧
1413, 1wa 97 . . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1514, 4wex 1381 . . . . 5 wff 𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1615, 3wex 1381 . . . 4 wff 𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1716, 2wex 1381 . . 3 wff 𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
1817, 6cab 2026 . 2 class {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
195, 18wceq 1243 1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
Colors of variables: wff set class
This definition is referenced by:  oprabid  5515  dfoprab2  5530  nfoprab1  5532  nfoprab2  5533  nfoprab3  5534  nfoprab  5535  oprabbid  5536  ssoprab2  5539  mpt20  5552  cbvoprab2  5555  eloprabga  5569  oprabrexex2  5735  eloprabi  5800  dftpos3  5855
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