Step |
Hyp |
Ref |
Expression |
1 |
|
meetdmss.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
meetdmss.j |
⊢ ∧ = ( meet ‘ 𝐾 ) |
3 |
|
meetdmss.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
4 |
|
relopabv |
⊢ Rel { 〈 𝑥 , 𝑦 〉 ∣ { 𝑥 , 𝑦 } ∈ dom ( glb ‘ 𝐾 ) } |
5 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
6 |
5 2
|
meetdm |
⊢ ( 𝐾 ∈ 𝑉 → 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 |
⊢ ( 𝜑 → 𝑥 ∈ V ) |
12 |
|
vex |
⊢ 𝑦 ∈ V |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝑦 ∈ 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 ) ) → 𝐾 ∈ 𝑉 ) |
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 |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ↔ { 𝑥 , 𝑦 } ⊆ 𝐵 ) |
21 |
|
opelxpi |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 〈 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 ) ) |