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Definition df-md 29123
Description: Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 29137 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
df-md 𝑀 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))))}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-md
StepHypRef Expression
1 cmd 27807 . 2 class 𝑀
2 vx . . . . . . 7 setvar 𝑥
32cv 1481 . . . . . 6 class 𝑥
4 cch 27770 . . . . . 6 class C
53, 4wcel 1989 . . . . 5 wff 𝑥C
6 vy . . . . . . 7 setvar 𝑦
76cv 1481 . . . . . 6 class 𝑦
87, 4wcel 1989 . . . . 5 wff 𝑦C
95, 8wa 384 . . . 4 wff (𝑥C𝑦C )
10 vz . . . . . . . 8 setvar 𝑧
1110cv 1481 . . . . . . 7 class 𝑧
1211, 7wss 3572 . . . . . 6 wff 𝑧𝑦
13 chj 27774 . . . . . . . . 9 class
1411, 3, 13co 6647 . . . . . . . 8 class (𝑧 𝑥)
1514, 7cin 3571 . . . . . . 7 class ((𝑧 𝑥) ∩ 𝑦)
163, 7cin 3571 . . . . . . . 8 class (𝑥𝑦)
1711, 16, 13co 6647 . . . . . . 7 class (𝑧 (𝑥𝑦))
1815, 17wceq 1482 . . . . . 6 wff ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))
1912, 18wi 4 . . . . 5 wff (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦)))
2019, 10, 4wral 2911 . . . 4 wff 𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦)))
219, 20wa 384 . . 3 wff ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))))
2221, 2, 6copab 4710 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))))}
231, 22wceq 1482 1 wff 𝑀 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑧𝑦 → ((𝑧 𝑥) ∩ 𝑦) = (𝑧 (𝑥𝑦))))}
Colors of variables: wff setvar class
This definition is referenced by:  mdbr  29137
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