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Definition df-opab 4067
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 4065 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 vz . . . . . . . 8 setvar 𝑧
65cv 1352 . . . . . . 7 class 𝑧
72cv 1352 . . . . . . . 8 class 𝑥
83cv 1352 . . . . . . . 8 class 𝑦
97, 8cop 3597 . . . . . . 7 class 𝑥, 𝑦
106, 9wceq 1353 . . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦
1110, 1wa 104 . . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1211, 3wex 1492 . . . 4 wff 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1312, 2wex 1492 . . 3 wff 𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1413, 5cab 2163 . 2 class {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
154, 14wceq 1353 1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Colors of variables: wff set class
This definition is referenced by:  opabss  4069  opabbid  4070  nfopab  4073  nfopab1  4074  nfopab2  4075  cbvopab  4076  cbvopab1  4078  cbvopab2  4079  cbvopab1s  4080  cbvopab2v  4082  unopab  4084  opabid  4259  elopab  4260  ssopab2  4277  iunopab  4283  elxpi  4644  rabxp  4665  csbxpg  4709  relopabi  4754  opabbrex  5922  dfoprab2  5925  dmoprab  5959  dfopab2  6193  cnvoprab  6238
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