Metamath Proof Explorer


Theorem meetdmss

Description: Subset property of domain of meet. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses meetdmss.b 𝐵 = ( Base ‘ 𝐾 )
meetdmss.j = ( meet ‘ 𝐾 )
meetdmss.k ( 𝜑𝐾𝑉 )
Assertion meetdmss ( 𝜑 → dom ⊆ ( 𝐵 × 𝐵 ) )

Proof

Step Hyp Ref Expression
1 meetdmss.b 𝐵 = ( Base ‘ 𝐾 )
2 meetdmss.j = ( meet ‘ 𝐾 )
3 meetdmss.k ( 𝜑𝐾𝑉 )
4 relopabv Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) }
5 eqid ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
6 5 2 meetdm ( 𝐾𝑉 → dom = { ⟨ 𝑥 , 𝑦 ⟩ ∣ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) } )
7 3 6 syl ( 𝜑 → dom = { ⟨ 𝑥 , 𝑦 ⟩ ∣ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) } )
8 7 releqd ( 𝜑 → ( Rel dom ↔ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) } ) )
9 4 8 mpbiri ( 𝜑 → Rel dom )
10 vex 𝑥 ∈ V
11 10 a1i ( 𝜑𝑥 ∈ V )
12 vex 𝑦 ∈ V
13 12 a1i ( 𝜑𝑦 ∈ V )
14 5 2 3 11 13 meetdef ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ dom ↔ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) ) )
15 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
16 3 adantr ( ( 𝜑 ∧ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) ) → 𝐾𝑉 )
17 simpr ( ( 𝜑 ∧ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) ) → { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) )
18 1 15 5 16 17 glbelss ( ( 𝜑 ∧ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) ) → { 𝑥 , 𝑦 } ⊆ 𝐵 )
19 18 ex ( 𝜑 → ( { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) → { 𝑥 , 𝑦 } ⊆ 𝐵 ) )
20 10 12 prss ( ( 𝑥𝐵𝑦𝐵 ) ↔ { 𝑥 , 𝑦 } ⊆ 𝐵 )
21 opelxpi ( ( 𝑥𝐵𝑦𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐵 × 𝐵 ) )
22 20 21 sylbir ( { 𝑥 , 𝑦 } ⊆ 𝐵 → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐵 × 𝐵 ) )
23 19 22 syl6 ( 𝜑 → ( { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐵 × 𝐵 ) ) )
24 14 23 sylbid ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ dom → ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐵 × 𝐵 ) ) )
25 9 24 relssdv ( 𝜑 → dom ⊆ ( 𝐵 × 𝐵 ) )