ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-opab GIF version

Definition df-opab 4065
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 4063 . 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 3595 . . . . . . 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  4067  opabbid  4068  nfopab  4071  nfopab1  4072  nfopab2  4073  cbvopab  4074  cbvopab1  4076  cbvopab2  4077  cbvopab1s  4078  cbvopab2v  4080  unopab  4082  opabid  4257  elopab  4258  ssopab2  4275  iunopab  4281  elxpi  4642  rabxp  4663  csbxpg  4707  relopabi  4752  opabbrex  5918  dfoprab2  5921  dmoprab  5955  dfopab2  6189  cnvoprab  6234
  Copyright terms: Public domain W3C validator