Definition df-oprab 7402

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 7401 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 7558. (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 7399	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6		vw	. . . . . . . . 9 setvar 𝑤
7	6	cv 1561	. . . . . . . 8 class 𝑤
8	2	cv 1561	. . . . . . . . . 10 class 𝑥
9	3	cv 1561	. . . . . . . . . 10 class 𝑦
10	8, 9	cop 4590	. . . . . . . . 9 class ⟨𝑥, 𝑦⟩
11	4	cv 1561	. . . . . . . . 9 class 𝑧
12	10, 11	cop 4590	. . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
13	7, 12	wceq 1562	. . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
14	13, 1	wa 399	. . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
15	14, 4	wex 1801	. . . . 5 wff ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
16	15, 3	wex 1801	. . . 4 wff ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
17	16, 2	wex 1801	. . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
18	17, 6	cab 2742	. 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19	5, 18	wceq 1562	1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}

This definition is referenced by:  oprabidw  7429  oprabid  7430  dfoprab2  7456  nfoprab1  7459  nfoprab2  7460  nfoprab3  7461  nfoprab  7462  oprabbid  7463  oprabbidv  7464  ssoprab2  7466  0mpo0  7481  mpo0  7483  cbvoprab2  7486  cbvoprab12v  7488  cbvoprab3v  7490  eloprabga  7507  oprabrexex2  7961  eloprabi  8046  dftpos3  8226  join0  18437  meet0  18438  mppspstlem  35926  mppsval  35927  colinearex  36415  cbvoprab1vw  36602  cbvoprab2vw  36603  cbvoprab123vw  36604  cbvoprab23vw  36605  cbvoprab13vw  36606  cbvoprab1davw  36636  cbvoprab2davw  36637  cbvoprab3davw  36638  cbvoprab123davw  36639  cbvoprab12davw  36640  cbvoprab23davw  36641  cbvoprab13davw  36642  csboprabg  37829  eloprab1st2nd  49494