Definition df-oprab 5845

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 5844 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 5973. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
df-oprab	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(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

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 5842	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6		vw	. . . . . . . . 9 setvar 𝑤
7	6	cv 1342	. . . . . . . 8 class 𝑤
8	2	cv 1342	. . . . . . . . . 10 class 𝑥
9	3	cv 1342	. . . . . . . . . 10 class 𝑦
10	8, 9	cop 3578	. . . . . . . . 9 class ⟨𝑥, 𝑦⟩
11	4	cv 1342	. . . . . . . . 9 class 𝑧
12	10, 11	cop 3578	. . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
13	7, 12	wceq 1343	. . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
14	13, 1	wa 103	. . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
15	14, 4	wex 1480	. . . . 5 wff ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
16	15, 3	wex 1480	. . . 4 wff ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
17	16, 2	wex 1480	. . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
18	17, 6	cab 2151	. 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19	5, 18	wceq 1343	1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}

This definition is referenced by:  oprabid  5870  dfoprab2  5885  nfoprab1  5887  nfoprab2  5888  nfoprab3  5889  nfoprab  5890  oprabbid  5891  ssoprab2  5894  mpo0  5908  cbvoprab2  5911  eloprabga  5925  oprabrexex2  6095  eloprabi  6161  cnvoprab  6198  dftpos3  6226