Definition df-opab 4092

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 4090	. 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 3622	. . . . . . 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 1503	. . . 4 wff ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
13	12, 2	wex 1503	. . 3 wff ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
14	13, 5	cab 2179	. 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  4094  opabbid  4095  nfopab  4098  nfopab1  4099  nfopab2  4100  cbvopab  4101  cbvopab1  4103  cbvopab2  4104  cbvopab1s  4105  cbvopab2v  4107  unopab  4109  opabid  4287  elopab  4289  ssopab2  4307  iunopab  4313  elxpi  4676  rabxp  4697  csbxpg  4741  relopabi  4788  opabbrex  5963  dfoprab2  5966  dmoprab  6000  dfopab2  6244  cnvoprab  6289