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Definition df-opab 4027
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 ∣ ∃𝑥𝑦(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
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
41, 2, 3copab 4025 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 vz . . . . . . . 8 setvar 𝑧
65cv 1334 . . . . . . 7 class 𝑧
72cv 1334 . . . . . . . 8 class 𝑥
83cv 1334 . . . . . . . 8 class 𝑦
97, 8cop 3563 . . . . . . 7 class 𝑥, 𝑦
106, 9wceq 1335 . . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦
1110, 1wa 103 . . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1211, 3wex 1472 . . . 4 wff 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1312, 2wex 1472 . . 3 wff 𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1413, 5cab 2143 . 2 class {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
154, 14wceq 1335 1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Colors of variables: wff set class
This definition is referenced by:  opabss  4029  opabbid  4030  nfopab  4033  nfopab1  4034  nfopab2  4035  cbvopab  4036  cbvopab1  4038  cbvopab2  4039  cbvopab1s  4040  cbvopab2v  4042  unopab  4044  opabid  4218  elopab  4219  ssopab2  4236  iunopab  4242  elxpi  4603  rabxp  4624  csbxpg  4668  relopabi  4713  opabbrex  5866  dfoprab2  5869  dmoprab  5903  dfopab2  6138  cnvoprab  6182
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