Step |
Hyp |
Ref |
Expression |
1 |
|
ordtNEW.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ordtNEW.l |
⊢ ≤ = ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) |
3 |
|
ordtrest2NEW.2 |
⊢ ( 𝜑 → 𝐾 ∈ Toset ) |
4 |
|
ordtrest2NEW.3 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
5 |
|
ordtrest2NEW.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → { 𝑧 ∈ 𝐵 ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } ⊆ 𝐴 ) |
6 |
|
tospos |
⊢ ( 𝐾 ∈ Toset → 𝐾 ∈ Poset ) |
7 |
|
posprs |
⊢ ( 𝐾 ∈ Poset → 𝐾 ∈ Proset ) |
8 |
3 6 7
|
3syl |
⊢ ( 𝜑 → 𝐾 ∈ Proset ) |
9 |
1 2
|
ordtrestNEW |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ⊆ ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) ) |
10 |
8 4 9
|
syl2anc |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ⊆ ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) ) |
11 |
|
eqid |
⊢ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) = ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) |
12 |
|
eqid |
⊢ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) = ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) |
13 |
1 2 11 12
|
ordtprsval |
⊢ ( 𝐾 ∈ Proset → ( ordTop ‘ ≤ ) = ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ) ) |
14 |
8 13
|
syl |
⊢ ( 𝜑 → ( ordTop ‘ ≤ ) = ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ) ) |
15 |
14
|
oveq1d |
⊢ ( 𝜑 → ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) = ( ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ) ↾t 𝐴 ) ) |
16 |
|
fibas |
⊢ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ∈ TopBases |
17 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
18 |
17
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
19 |
18 4
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
20 |
|
tgrest |
⊢ ( ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ∈ TopBases ∧ 𝐴 ∈ V ) → ( topGen ‘ ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ↾t 𝐴 ) ) = ( ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ) ↾t 𝐴 ) ) |
21 |
16 19 20
|
sylancr |
⊢ ( 𝜑 → ( topGen ‘ ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ↾t 𝐴 ) ) = ( ( topGen ‘ ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ) ↾t 𝐴 ) ) |
22 |
15 21
|
eqtr4d |
⊢ ( 𝜑 → ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) = ( topGen ‘ ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ↾t 𝐴 ) ) ) |
23 |
|
firest |
⊢ ( fi ‘ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ) = ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ↾t 𝐴 ) |
24 |
23
|
fveq2i |
⊢ ( topGen ‘ ( fi ‘ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ) ) = ( topGen ‘ ( ( fi ‘ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) ↾t 𝐴 ) ) |
25 |
22 24
|
eqtr4di |
⊢ ( 𝜑 → ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) = ( topGen ‘ ( fi ‘ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ) ) ) |
26 |
|
fvex |
⊢ ( le ‘ 𝐾 ) ∈ V |
27 |
26
|
inex1 |
⊢ ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) ∈ V |
28 |
2 27
|
eqeltri |
⊢ ≤ ∈ V |
29 |
28
|
inex1 |
⊢ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ∈ V |
30 |
|
ordttop |
⊢ ( ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ∈ V → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ Top ) |
31 |
29 30
|
mp1i |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ Top ) |
32 |
1 2 11 12
|
ordtprsuni |
⊢ ( 𝐾 ∈ Proset → 𝐵 = ∪ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) |
33 |
8 32
|
syl |
⊢ ( 𝜑 → 𝐵 = ∪ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ) |
34 |
33 18
|
eqeltrrd |
⊢ ( 𝜑 → ∪ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ∈ V ) |
35 |
|
uniexb |
⊢ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ∈ V ↔ ∪ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ∈ V ) |
36 |
34 35
|
sylibr |
⊢ ( 𝜑 → ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ∈ V ) |
37 |
|
restval |
⊢ ( ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ∈ V ∧ 𝐴 ∈ V ) → ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) = ran ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) ) |
38 |
36 19 37
|
syl2anc |
⊢ ( 𝜑 → ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) = ran ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) ) |
39 |
|
sseqin2 |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |
40 |
4 39
|
sylib |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |
41 |
|
eqid |
⊢ dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) |
42 |
41
|
ordttopon |
⊢ ( ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ∈ V → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ ( TopOn ‘ dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
43 |
29 42
|
mp1i |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ ( TopOn ‘ dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
44 |
1 2
|
prsssdm |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = 𝐴 ) |
45 |
8 4 44
|
syl2anc |
⊢ ( 𝜑 → dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = 𝐴 ) |
46 |
45
|
fveq2d |
⊢ ( 𝜑 → ( TopOn ‘ dom ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( TopOn ‘ 𝐴 ) ) |
47 |
43 46
|
eleqtrd |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ ( TopOn ‘ 𝐴 ) ) |
48 |
|
toponmax |
⊢ ( ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ ( TopOn ‘ 𝐴 ) → 𝐴 ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
49 |
47 48
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
50 |
40 49
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
51 |
|
elsni |
⊢ ( 𝑣 ∈ { 𝐵 } → 𝑣 = 𝐵 ) |
52 |
51
|
ineq1d |
⊢ ( 𝑣 ∈ { 𝐵 } → ( 𝑣 ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) ) |
53 |
52
|
eleq1d |
⊢ ( 𝑣 ∈ { 𝐵 } → ( ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ( 𝐵 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
54 |
50 53
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑣 ∈ { 𝐵 } → ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
55 |
54
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑣 ∈ { 𝐵 } ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
56 |
1 2 3 4 5
|
ordtrest2NEWlem |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
57 |
|
eqid |
⊢ ( ODual ‘ 𝐾 ) = ( ODual ‘ 𝐾 ) |
58 |
57 1
|
odubas |
⊢ 𝐵 = ( Base ‘ ( ODual ‘ 𝐾 ) ) |
59 |
2
|
cnveqi |
⊢ ◡ ≤ = ◡ ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) |
60 |
|
cnvin |
⊢ ◡ ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) = ( ◡ ( le ‘ 𝐾 ) ∩ ◡ ( 𝐵 × 𝐵 ) ) |
61 |
|
cnvxp |
⊢ ◡ ( 𝐵 × 𝐵 ) = ( 𝐵 × 𝐵 ) |
62 |
61
|
ineq2i |
⊢ ( ◡ ( le ‘ 𝐾 ) ∩ ◡ ( 𝐵 × 𝐵 ) ) = ( ◡ ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) |
63 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
64 |
57 63
|
oduleval |
⊢ ◡ ( le ‘ 𝐾 ) = ( le ‘ ( ODual ‘ 𝐾 ) ) |
65 |
64
|
ineq1i |
⊢ ( ◡ ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) = ( ( le ‘ ( ODual ‘ 𝐾 ) ) ∩ ( 𝐵 × 𝐵 ) ) |
66 |
60 62 65
|
3eqtri |
⊢ ◡ ( ( le ‘ 𝐾 ) ∩ ( 𝐵 × 𝐵 ) ) = ( ( le ‘ ( ODual ‘ 𝐾 ) ) ∩ ( 𝐵 × 𝐵 ) ) |
67 |
59 66
|
eqtri |
⊢ ◡ ≤ = ( ( le ‘ ( ODual ‘ 𝐾 ) ) ∩ ( 𝐵 × 𝐵 ) ) |
68 |
57
|
odutos |
⊢ ( 𝐾 ∈ Toset → ( ODual ‘ 𝐾 ) ∈ Toset ) |
69 |
3 68
|
syl |
⊢ ( 𝜑 → ( ODual ‘ 𝐾 ) ∈ Toset ) |
70 |
|
vex |
⊢ 𝑦 ∈ V |
71 |
|
vex |
⊢ 𝑧 ∈ V |
72 |
70 71
|
brcnv |
⊢ ( 𝑦 ◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦 ) |
73 |
|
vex |
⊢ 𝑥 ∈ V |
74 |
71 73
|
brcnv |
⊢ ( 𝑧 ◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧 ) |
75 |
72 74
|
anbi12ci |
⊢ ( ( 𝑦 ◡ ≤ 𝑧 ∧ 𝑧 ◡ ≤ 𝑥 ) ↔ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) ) |
76 |
75
|
rabbii |
⊢ { 𝑧 ∈ 𝐵 ∣ ( 𝑦 ◡ ≤ 𝑧 ∧ 𝑧 ◡ ≤ 𝑥 ) } = { 𝑧 ∈ 𝐵 ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } |
77 |
76 5
|
eqsstrid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → { 𝑧 ∈ 𝐵 ∣ ( 𝑦 ◡ ≤ 𝑧 ∧ 𝑧 ◡ ≤ 𝑥 ) } ⊆ 𝐴 ) |
78 |
77
|
ancom2s |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) → { 𝑧 ∈ 𝐵 ∣ ( 𝑦 ◡ ≤ 𝑧 ∧ 𝑧 ◡ ≤ 𝑥 ) } ⊆ 𝐴 ) |
79 |
58 67 69 4 78
|
ordtrest2NEWlem |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ◡ ≤ 𝑧 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
80 |
|
vex |
⊢ 𝑤 ∈ V |
81 |
80 71
|
brcnv |
⊢ ( 𝑤 ◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑤 ) |
82 |
81
|
bicomi |
⊢ ( 𝑧 ≤ 𝑤 ↔ 𝑤 ◡ ≤ 𝑧 ) |
83 |
82
|
a1i |
⊢ ( 𝜑 → ( 𝑧 ≤ 𝑤 ↔ 𝑤 ◡ ≤ 𝑧 ) ) |
84 |
83
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝑧 ≤ 𝑤 ↔ ¬ 𝑤 ◡ ≤ 𝑧 ) ) |
85 |
84
|
rabbidv |
⊢ ( 𝜑 → { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } = { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ◡ ≤ 𝑧 } ) |
86 |
85
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) = ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ◡ ≤ 𝑧 } ) ) |
87 |
86
|
rneqd |
⊢ ( 𝜑 → ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) = ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ◡ ≤ 𝑧 } ) ) |
88 |
|
cnvin |
⊢ ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ◡ ≤ ∩ ◡ ( 𝐴 × 𝐴 ) ) |
89 |
|
cnvxp |
⊢ ◡ ( 𝐴 × 𝐴 ) = ( 𝐴 × 𝐴 ) |
90 |
89
|
ineq2i |
⊢ ( ◡ ≤ ∩ ◡ ( 𝐴 × 𝐴 ) ) = ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) |
91 |
88 90
|
eqtri |
⊢ ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) |
92 |
91
|
fveq2i |
⊢ ( ordTop ‘ ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ordTop ‘ ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) ) |
93 |
1
|
ressprs |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐾 ↾s 𝐴 ) ∈ Proset ) |
94 |
8 4 93
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ↾s 𝐴 ) ∈ Proset ) |
95 |
|
eqid |
⊢ ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) |
96 |
|
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 ) ) ) ) |
97 |
95 96
|
ordtcnvNEW |
⊢ ( ( 𝐾 ↾s 𝐴 ) ∈ Proset → ( ordTop ‘ ◡ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) = ( ordTop ‘ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) ) |
98 |
94 97
|
syl |
⊢ ( 𝜑 → ( ordTop ‘ ◡ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) = ( ordTop ‘ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) ) |
99 |
1 2
|
prsss |
⊢ ( ( 𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵 ) → ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) ) |
100 |
8 4 99
|
syl2anc |
⊢ ( 𝜑 → ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) ) |
101 |
|
eqid |
⊢ ( 𝐾 ↾s 𝐴 ) = ( 𝐾 ↾s 𝐴 ) |
102 |
101 63
|
ressle |
⊢ ( 𝐴 ∈ V → ( le ‘ 𝐾 ) = ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
103 |
19 102
|
syl |
⊢ ( 𝜑 → ( le ‘ 𝐾 ) = ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
104 |
101 1
|
ressbas2 |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
105 |
4 104
|
syl |
⊢ ( 𝜑 → 𝐴 = ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) |
106 |
105
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝐴 × 𝐴 ) = ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) |
107 |
103 106
|
ineq12d |
⊢ ( 𝜑 → ( ( le ‘ 𝐾 ) ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) |
108 |
100 107
|
eqtrd |
⊢ ( 𝜑 → ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) |
109 |
108
|
cnveqd |
⊢ ( 𝜑 → ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) = ◡ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) |
110 |
109
|
fveq2d |
⊢ ( 𝜑 → ( ordTop ‘ ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ordTop ‘ ◡ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) ) |
111 |
108
|
fveq2d |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ordTop ‘ ( ( le ‘ ( 𝐾 ↾s 𝐴 ) ) ∩ ( ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) × ( Base ‘ ( 𝐾 ↾s 𝐴 ) ) ) ) ) ) |
112 |
98 110 111
|
3eqtr4d |
⊢ ( 𝜑 → ( ordTop ‘ ◡ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
113 |
92 112
|
syl5reqr |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ordTop ‘ ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
114 |
113
|
eleq2d |
⊢ ( 𝜑 → ( ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
115 |
87 114
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ◡ ≤ 𝑧 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ◡ ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
116 |
79 115
|
mpbird |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
117 |
|
ralunb |
⊢ ( ∀ 𝑣 ∈ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ( ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∧ ∀ 𝑣 ∈ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
118 |
56 116 117
|
sylanbrc |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
119 |
|
ralunb |
⊢ ( ∀ 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ( ∀ 𝑣 ∈ { 𝐵 } ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∧ ∀ 𝑣 ∈ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) ) |
120 |
55 118 119
|
sylanbrc |
⊢ ( 𝜑 → ∀ 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
121 |
|
eqid |
⊢ ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) = ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) |
122 |
121
|
fmpt |
⊢ ( ∀ 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ( 𝑣 ∩ 𝐴 ) ∈ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ↔ ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) : ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ⟶ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
123 |
120 122
|
sylib |
⊢ ( 𝜑 → ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) : ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ⟶ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
124 |
123
|
frnd |
⊢ ( 𝜑 → ran ( 𝑣 ∈ ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↦ ( 𝑣 ∩ 𝐴 ) ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
125 |
38 124
|
eqsstrd |
⊢ ( 𝜑 → ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
126 |
|
tgfiss |
⊢ ( ( ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ∈ Top ∧ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) → ( topGen ‘ ( fi ‘ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ) ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
127 |
31 125 126
|
syl2anc |
⊢ ( 𝜑 → ( topGen ‘ ( fi ‘ ( ( { 𝐵 } ∪ ( ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧 } ) ∪ ran ( 𝑧 ∈ 𝐵 ↦ { 𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤 } ) ) ) ↾t 𝐴 ) ) ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
128 |
25 127
|
eqsstrd |
⊢ ( 𝜑 → ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) ⊆ ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) ) |
129 |
10 128
|
eqssd |
⊢ ( 𝜑 → ( ordTop ‘ ( ≤ ∩ ( 𝐴 × 𝐴 ) ) ) = ( ( ordTop ‘ ≤ ) ↾t 𝐴 ) ) |