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Definition df-opab 4051
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 4049 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 vz . . . . . . . 8 setvar 𝑧
65cv 1347 . . . . . . 7 class 𝑧
72cv 1347 . . . . . . . 8 class 𝑥
83cv 1347 . . . . . . . 8 class 𝑦
97, 8cop 3586 . . . . . . 7 class 𝑥, 𝑦
106, 9wceq 1348 . . . . . 6 wff 𝑧 = ⟨𝑥, 𝑦
1110, 1wa 103 . . . . 5 wff (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1211, 3wex 1485 . . . 4 wff 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1312, 2wex 1485 . . 3 wff 𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1413, 5cab 2156 . 2 class {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
154, 14wceq 1348 1 wff {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Colors of variables: wff set class
This definition is referenced by:  opabss  4053  opabbid  4054  nfopab  4057  nfopab1  4058  nfopab2  4059  cbvopab  4060  cbvopab1  4062  cbvopab2  4063  cbvopab1s  4064  cbvopab2v  4066  unopab  4068  opabid  4242  elopab  4243  ssopab2  4260  iunopab  4266  elxpi  4627  rabxp  4648  csbxpg  4692  relopabi  4737  opabbrex  5897  dfoprab2  5900  dmoprab  5934  dfopab2  6168  cnvoprab  6213
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