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Definition df-opab 4066
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 4064 . 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 3596 . . . . . . 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  4068  opabbid  4069  nfopab  4072  nfopab1  4073  nfopab2  4074  cbvopab  4075  cbvopab1  4077  cbvopab2  4078  cbvopab1s  4079  cbvopab2v  4081  unopab  4083  opabid  4258  elopab  4259  ssopab2  4276  iunopab  4282  elxpi  4643  rabxp  4664  csbxpg  4708  relopabi  4753  opabbrex  5919  dfoprab2  5922  dmoprab  5956  dfopab2  6190  cnvoprab  6235
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