Step |
Hyp |
Ref |
Expression |
1 |
|
ordtNEW.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ordtNEW.l |
⊢ ≤ = ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) |
3 |
1 2
|
prsss |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) ) |
4 |
3
|
dmeqd |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = dom ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) ) |
5 |
1
|
ressprs |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐾 ↾s 𝐴 ) ∈ Proset ) |
6 |
|
eqid |
⊢ ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) |
7 |
|
eqid |
⊢ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) = ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) |
8 |
6 7
|
prsdm |
⊢ ( ( 𝐾 ↾s 𝐴 ) ∈ Proset → dom ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
9 |
5 8
|
syl |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
10 |
|
eqid |
⊢ ( 𝐾 ↾s 𝐴 ) = ( 𝐾 ↾s 𝐴 ) |
11 |
10 1
|
ressbas2 |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
12 |
|
fvex |
⊢ ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ∈ V |
13 |
11 12
|
eqeltrdi |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V ) |
14 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
15 |
10 14
|
ressle |
⊢ ( 𝐴 ∈ V → ( le ‘ 𝐾 ) = ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
16 |
13 15
|
syl |
⊢ ( 𝐴 ⊆ 𝐵 → ( le ‘ 𝐾 ) = ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( le ‘ 𝐾 ) = ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
18 |
11
|
adantl |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
19 |
18
|
sqxpeqd |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 × 𝐴 ) = ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) |
20 |
17 19
|
ineq12d |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) |
21 |
20
|
dmeqd |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) = dom ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) |
22 |
9 21 18
|
3eqtr4d |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) = 𝐴 ) |
23 |
4 22
|
eqtrd |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = 𝐴 ) |