Definition df-oprab 3004

Description: Define class abstraction of nested ordered pairs (for use in defining operations). A special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally x, y, and z are distinct, although the definition doesn't strictly require it.

Assertion

Ref	Expression
df-oprab	⊢ {⟨⟨x, y⟩, z⟩∣φ} = {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}

Distinct variable group(s):   x,w   y,w   z,w   φ,w

Detailed syntax breakdown of Definition df-oprab

Step	Hyp	Ref	Expression
1		wph	. . 3 wff φ
2		vx	. . 3 set x
3		vy	. . 3 set y
4		vz	. . 3 set z
5	1, 2, 3, 4	copab2 3002	. 2 class {⟨⟨x, y⟩, z⟩∣φ}
6		vw	. . . . . . . . 9 set w
7	6	cv 1089	. . . . . . . 8 class w
8	2	cv 1089	. . . . . . . . . 10 class x
9	3	cv 1089	. . . . . . . . . 10 class y
10	8, 9	cop 1810	. . . . . . . . 9 class ⟨x, y⟩
11	4	cv 1089	. . . . . . . . 9 class z
12	10, 11	cop 1810	. . . . . . . 8 class ⟨⟨x, y⟩, z⟩
13	7, 12	wceq 1091	. . . . . . 7 wff w = ⟨⟨x, y⟩, z⟩
14	13, 1	wa 196	. . . . . 6 wff (w = ⟨⟨x, y⟩, z⟩ ∧ φ)
15	14, 4	wex 678	. . . . 5 wff ∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)
16	15, 3	wex 678	. . . 4 wff ∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)
17	16, 2	wex 678	. . 3 wff ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)
18	17, 6	cab 1090	. 2 class {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
19	5, 18	wceq 1091	1 wff {⟨⟨x, y⟩, z⟩∣φ} = {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}

This definition is referenced by:  dfoprab2 3021  hboprab1 3023  hboprab2 3024  cbvoprab12 3028  eloprabg 3035

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