Step | Hyp | Ref
| Expression |
1 | | zex 11386 |
. . . . . 6
⊢ ℤ
∈ V |
2 | 1 | pwex 4848 |
. . . . 5
⊢ 𝒫
ℤ ∈ V |
3 | | uzf 11690 |
. . . . . 6
⊢
ℤ≥:ℤ⟶𝒫 ℤ |
4 | | frn 6053 |
. . . . . 6
⊢
(ℤ≥:ℤ⟶𝒫 ℤ → ran
ℤ≥ ⊆ 𝒫 ℤ) |
5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢ ran
ℤ≥ ⊆ 𝒫 ℤ |
6 | 2, 5 | ssexi 4803 |
. . . 4
⊢ ran
ℤ≥ ∈ V |
7 | | uzfbas.1 |
. . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) |
8 | | fvex 6201 |
. . . . 5
⊢
(ℤ≥‘𝑀) ∈ V |
9 | 7, 8 | eqeltri 2697 |
. . . 4
⊢ 𝑍 ∈ V |
10 | | restval 16087 |
. . . 4
⊢ ((ran
ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran
ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍))) |
11 | 6, 9, 10 | mp2an 708 |
. . 3
⊢ (ran
ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍)) |
12 | 7 | ineq2i 3811 |
. . . . . . . . 9
⊢
((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩
(ℤ≥‘𝑀)) |
13 | | uzin 11720 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) =
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
14 | 13 | ancoms 469 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) =
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
15 | 12, 14 | syl5eq 2668 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
16 | | ffn 6045 |
. . . . . . . . . . 11
⊢
(ℤ≥:ℤ⟶𝒫 ℤ →
ℤ≥ Fn ℤ) |
17 | 3, 16 | ax-mp 5 |
. . . . . . . . . 10
⊢
ℤ≥ Fn ℤ |
18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
ℤ≥ Fn ℤ) |
19 | | uzssz 11707 |
. . . . . . . . . . 11
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
20 | 7, 19 | eqsstri 3635 |
. . . . . . . . . 10
⊢ 𝑍 ⊆
ℤ |
21 | 20 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑍 ⊆
ℤ) |
22 | | inss2 3834 |
. . . . . . . . . 10
⊢
((ℤ≥‘𝑦) ∩ 𝑍) ⊆ 𝑍 |
23 | | ifcl 4130 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ) |
24 | | uzid 11702 |
. . . . . . . . . . . 12
⊢ (if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
26 | 25, 15 | eleqtrrd 2704 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍)) |
27 | 22, 26 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍) |
28 | | fnfvima 6496 |
. . . . . . . . 9
⊢
((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍) →
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “
𝑍)) |
29 | 18, 21, 27, 28 | syl3anc 1326 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “
𝑍)) |
30 | 15, 29 | eqeltrd 2701 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
31 | 30 | ralrimiva 2966 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
32 | | ineq1 3807 |
. . . . . . . . 9
⊢ (𝑥 =
(ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍)) |
33 | 32 | eleq1d 2686 |
. . . . . . . 8
⊢ (𝑥 =
(ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍))) |
34 | 33 | ralrn 6362 |
. . . . . . 7
⊢
(ℤ≥ Fn ℤ → (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ ∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍))) |
35 | 17, 34 | ax-mp 5 |
. . . . . 6
⊢
(∀𝑥 ∈
ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ ∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
36 | 31, 35 | sylibr 224 |
. . . . 5
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ ran
ℤ≥(𝑥
∩ 𝑍) ∈
(ℤ≥ “ 𝑍)) |
37 | | eqid 2622 |
. . . . . 6
⊢ (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍)) |
38 | 37 | fmpt 6381 |
. . . . 5
⊢
(∀𝑥 ∈
ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran
ℤ≥⟶(ℤ≥ “ 𝑍)) |
39 | 36, 38 | sylib 208 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran
ℤ≥⟶(ℤ≥ “ 𝑍)) |
40 | | frn 6053 |
. . . 4
⊢ ((𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran
ℤ≥⟶(ℤ≥ “ 𝑍) → ran (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “
𝑍)) |
41 | 39, 40 | syl 17 |
. . 3
⊢ (𝑀 ∈ ℤ → ran
(𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “
𝑍)) |
42 | 11, 41 | syl5eqss 3649 |
. 2
⊢ (𝑀 ∈ ℤ → (ran
ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “
𝑍)) |
43 | 7 | uztrn2 11705 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍) |
44 | 43 | ex 450 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍)) |
45 | 44 | ssrdv 3609 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍) |
46 | 45 | adantl 482 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍) |
47 | | df-ss 3588 |
. . . . . 6
⊢
((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥)) |
48 | 46, 47 | sylib 208 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) →
((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥)) |
49 | 6 | a1i 11 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ran ℤ≥ ∈
V) |
50 | 9 | a1i 11 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑍 ∈ V) |
51 | 20 | sseli 3599 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ) |
52 | 51 | adantl 482 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ) |
53 | | fnfvelrn 6356 |
. . . . . . 7
⊢
((ℤ≥ Fn ℤ ∧ 𝑥 ∈ ℤ) →
(ℤ≥‘𝑥) ∈ ran
ℤ≥) |
54 | 17, 52, 53 | sylancr 695 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ ran
ℤ≥) |
55 | | elrestr 16089 |
. . . . . 6
⊢ ((ran
ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧
(ℤ≥‘𝑥) ∈ ran ℤ≥) →
((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥
↾t 𝑍)) |
56 | 49, 50, 54, 55 | syl3anc 1326 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) →
((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥
↾t 𝑍)) |
57 | 48, 56 | eqeltrrd 2702 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran
ℤ≥ ↾t 𝑍)) |
58 | 57 | ralrimiva 2966 |
. . 3
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ 𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍)) |
59 | | ffun 6048 |
. . . . 5
⊢
(ℤ≥:ℤ⟶𝒫 ℤ → Fun
ℤ≥) |
60 | 3, 59 | ax-mp 5 |
. . . 4
⊢ Fun
ℤ≥ |
61 | 3 | fdmi 6052 |
. . . . 5
⊢ dom
ℤ≥ = ℤ |
62 | 20, 61 | sseqtr4i 3638 |
. . . 4
⊢ 𝑍 ⊆ dom
ℤ≥ |
63 | | funimass4 6247 |
. . . 4
⊢ ((Fun
ℤ≥ ∧ 𝑍 ⊆ dom ℤ≥) →
((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)
↔ ∀𝑥 ∈
𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍))) |
64 | 60, 62, 63 | mp2an 708 |
. . 3
⊢
((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)
↔ ∀𝑥 ∈
𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍)) |
65 | 58, 64 | sylibr 224 |
. 2
⊢ (𝑀 ∈ ℤ →
(ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)) |
66 | 42, 65 | eqssd 3620 |
1
⊢ (𝑀 ∈ ℤ → (ran
ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍)) |