Definition df-oprab 7374

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 7373 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 7530. (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 7371	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6		vw	. . . . . . . . 9 setvar 𝑤
7	6	cv 1541	. . . . . . . 8 class 𝑤
8	2	cv 1541	. . . . . . . . . 10 class 𝑥
9	3	cv 1541	. . . . . . . . . 10 class 𝑦
10	8, 9	cop 4588	. . . . . . . . 9 class ⟨𝑥, 𝑦⟩
11	4	cv 1541	. . . . . . . . 9 class 𝑧
12	10, 11	cop 4588	. . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
13	7, 12	wceq 1542	. . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
14	13, 1	wa 395	. . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
15	14, 4	wex 1781	. . . . 5 wff ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
16	15, 3	wex 1781	. . . 4 wff ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
17	16, 2	wex 1781	. . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
18	17, 6	cab 2715	. 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19	5, 18	wceq 1542	1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}

This definition is referenced by:  oprabidw  7401  oprabid  7402  dfoprab2  7428  nfoprab1  7431  nfoprab2  7432  nfoprab3  7433  nfoprab  7434  oprabbid  7435  oprabbidv  7436  ssoprab2  7438  0mpo0  7453  mpo0  7455  cbvoprab2  7458  cbvoprab12v  7460  cbvoprab3v  7462  eloprabga  7479  oprabrexex2  7934  eloprabi  8019  dftpos3  8198  join0  18340  meet0  18341  mppspstlem  35793  mppsval  35794  colinearex  36282  cbvoprab1vw  36459  cbvoprab2vw  36460  cbvoprab123vw  36461  cbvoprab23vw  36462  cbvoprab13vw  36463  cbvoprab1davw  36493  cbvoprab2davw  36494  cbvoprab3davw  36495  cbvoprab123davw  36496  cbvoprab12davw  36497  cbvoprab23davw  36498  cbvoprab13davw  36499  csboprabg  37612  eloprab1st2nd  49256