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Metamath Proof Explorer

Theorem joincl

Description: Closure of join of elements in the domain. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses joincl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
joincl.j ⊢ ∨ = ( join ‘ 𝐾 )
joincl.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
joincl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
joincl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
joincl.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
Assertion joincl ( 𝜑 → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 joincl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 joincl.j ⊢ ∨ = ( join ‘ 𝐾 )
3 joincl.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
4 joincl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 joincl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 joincl.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
7 eqid ⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
8 7 2 3 4 5 joinval ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) = ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) )
9 7 2 3 4 5 joindef ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ↔ { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) ) )
10 6 9 mpbid ⊢ ( 𝜑 → { 𝑋 , 𝑌 } ∈ dom ( lub ‘ 𝐾 ) )
11 1 7 3 10 lubcl ⊢ ( 𝜑 → ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ∈ 𝐵 )
12 8 11 eqeltrd ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )