Definition df-opab 4095

Description: Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
df-opab	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Detailed syntax breakdown of Definition df-opab

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4	1, 2, 3	copab 4093	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5		vz	. . . . . . . 8 setvar 𝑧
6	5	cv 1363	. . . . . . 7 class 𝑧
7	2	cv 1363	. . . . . . . 8 class 𝑥
8	3	cv 1363	. . . . . . . 8 class 𝑦
9	7, 8	cop 3625	. . . . . . 7 class ⟨𝑥, 𝑦⟩
10	6, 9	wceq 1364	. . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦⟩
11	10, 1	wa 104	. . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
12	11, 3	wex 1506	. . . 4 wff ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
13	12, 2	wex 1506	. . 3 wff ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
14	13, 5	cab 2182	. 2 class {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15	4, 14	wceq 1364	1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}

This definition is referenced by:  opabss  4097  opabbid  4098  nfopab  4101  nfopab1  4102  nfopab2  4103  cbvopab  4104  cbvopab1  4106  cbvopab2  4107  cbvopab1s  4108  cbvopab2v  4110  unopab  4112  opabid  4290  elopab  4292  ssopab2  4310  iunopab  4316  elxpi  4679  rabxp  4700  csbxpg  4744  relopabi  4791  opabbrex  5966  dfoprab2  5969  dmoprab  6003  dfopab2  6247  cnvoprab  6292