Definition df-oprab 7350

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 7349 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 7506. (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 7347	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
6		vw	. . . . . . . . 9 setvar 𝑤
7	6	cv 1540	. . . . . . . 8 class 𝑤
8	2	cv 1540	. . . . . . . . . 10 class 𝑥
9	3	cv 1540	. . . . . . . . . 10 class 𝑦
10	8, 9	cop 4579	. . . . . . . . 9 class ⟨𝑥, 𝑦⟩
11	4	cv 1540	. . . . . . . . 9 class 𝑧
12	10, 11	cop 4579	. . . . . . . 8 class ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
13	7, 12	wceq 1541	. . . . . . 7 wff 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
14	13, 1	wa 395	. . . . . 6 wff (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
15	14, 4	wex 1780	. . . . 5 wff ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
16	15, 3	wex 1780	. . . 4 wff ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
17	16, 2	wex 1780	. . 3 wff ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
18	17, 6	cab 2709	. 2 class {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19	5, 18	wceq 1541	1 wff {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}

This definition is referenced by:  oprabidw  7377  oprabid  7378  dfoprab2  7404  nfoprab1  7407  nfoprab2  7408  nfoprab3  7409  nfoprab  7410  oprabbid  7411  oprabbidv  7412  ssoprab2  7414  0mpo0  7429  mpo0  7431  cbvoprab2  7434  cbvoprab12v  7436  cbvoprab3v  7438  eloprabga  7455  oprabrexex2  7910  eloprabi  7995  dftpos3  8174  join0  18309  meet0  18310  mppspstlem  35615  mppsval  35616  colinearex  36102  cbvoprab1vw  36279  cbvoprab2vw  36280  cbvoprab123vw  36281  cbvoprab23vw  36282  cbvoprab13vw  36283  cbvoprab1davw  36313  cbvoprab2davw  36314  cbvoprab3davw  36315  cbvoprab123davw  36316  cbvoprab12davw  36317  cbvoprab23davw  36318  cbvoprab13davw  36319  csboprabg  37372  eloprab1st2nd  48907