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Definition df-join 17577
 Description: Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
df-join join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
Distinct variable group:   𝑥,𝑝,𝑦,𝑧

Detailed syntax breakdown of Definition df-join
StepHypRef Expression
1 cjn 17545 . 2 class join
2 vp . . 3 setvar 𝑝
3 cvv 3469 . . 3 class V
4 vx . . . . . . 7 setvar 𝑥
54cv 1537 . . . . . 6 class 𝑥
6 vy . . . . . . 7 setvar 𝑦
76cv 1537 . . . . . 6 class 𝑦
85, 7cpr 4541 . . . . 5 class {𝑥, 𝑦}
9 vz . . . . . 6 setvar 𝑧
109cv 1537 . . . . 5 class 𝑧
112cv 1537 . . . . . 6 class 𝑝
12 club 17543 . . . . . 6 class lub
1311, 12cfv 6334 . . . . 5 class (lub‘𝑝)
148, 10, 13wbr 5042 . . . 4 wff {𝑥, 𝑦} (lub‘𝑝)𝑧
1514, 4, 6, 9coprab 7141 . . 3 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧}
162, 3, 15cmpt 5122 . 2 class (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
171, 16wceq 1538 1 wff join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
 Colors of variables: wff setvar class This definition is referenced by:  joinfval  17602
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