Step | Hyp | Ref
| Expression |
1 | | zex 12185 |
. . . . . 6
⊢ ℤ
∈ V |
2 | 1 | pwex 5273 |
. . . . 5
⊢ 𝒫
ℤ ∈ V |
3 | | uzf 12441 |
. . . . . 6
⊢
ℤ≥:ℤ⟶𝒫 ℤ |
4 | | frn 6552 |
. . . . . 6
⊢
(ℤ≥:ℤ⟶𝒫 ℤ → ran
ℤ≥ ⊆ 𝒫 ℤ) |
5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢ ran
ℤ≥ ⊆ 𝒫 ℤ |
6 | 2, 5 | ssexi 5215 |
. . . 4
⊢ ran
ℤ≥ ∈ V |
7 | | uzfbas.1 |
. . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) |
8 | 7 | fvexi 6731 |
. . . 4
⊢ 𝑍 ∈ V |
9 | | restval 16931 |
. . . 4
⊢ ((ran
ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran
ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍))) |
10 | 6, 8, 9 | mp2an 692 |
. . 3
⊢ (ran
ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍)) |
11 | 7 | ineq2i 4124 |
. . . . . . . . 9
⊢
((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩
(ℤ≥‘𝑀)) |
12 | | uzin 12474 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) =
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
13 | 12 | ancoms 462 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) =
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
14 | 11, 13 | syl5eq 2790 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
15 | | ffn 6545 |
. . . . . . . . . 10
⊢
(ℤ≥:ℤ⟶𝒫 ℤ →
ℤ≥ Fn ℤ) |
16 | 3, 15 | ax-mp 5 |
. . . . . . . . 9
⊢
ℤ≥ Fn ℤ |
17 | | uzssz 12459 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
18 | 7, 17 | eqsstri 3935 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℤ |
19 | | ifcl 4484 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ) |
20 | | uzid 12453 |
. . . . . . . . . . . 12
⊢ (if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
21 | 19, 20 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦))) |
22 | 21, 14 | eleqtrrd 2841 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍)) |
23 | 22 | elin2d 4113 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍) |
24 | | fnfvima 7049 |
. . . . . . . . 9
⊢
((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍) →
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “
𝑍)) |
25 | 16, 18, 23, 24 | mp3an12i 1467 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
(ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “
𝑍)) |
26 | 14, 25 | eqeltrd 2838 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) →
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
27 | 26 | ralrimiva 3105 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
28 | | ineq1 4120 |
. . . . . . . . 9
⊢ (𝑥 =
(ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍)) |
29 | 28 | eleq1d 2822 |
. . . . . . . 8
⊢ (𝑥 =
(ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍))) |
30 | 29 | ralrn 6907 |
. . . . . . 7
⊢
(ℤ≥ Fn ℤ → (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ ∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍))) |
31 | 16, 30 | ax-mp 5 |
. . . . . 6
⊢
(∀𝑥 ∈
ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ ∀𝑦 ∈ ℤ
((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “
𝑍)) |
32 | 27, 31 | sylibr 237 |
. . . . 5
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ ran
ℤ≥(𝑥
∩ 𝑍) ∈
(ℤ≥ “ 𝑍)) |
33 | | eqid 2737 |
. . . . . 6
⊢ (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦
(𝑥 ∩ 𝑍)) |
34 | 33 | fmpt 6927 |
. . . . 5
⊢
(∀𝑥 ∈
ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “
𝑍) ↔ (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran
ℤ≥⟶(ℤ≥ “ 𝑍)) |
35 | 32, 34 | sylib 221 |
. . . 4
⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran
ℤ≥⟶(ℤ≥ “ 𝑍)) |
36 | 35 | frnd 6553 |
. . 3
⊢ (𝑀 ∈ ℤ → ran
(𝑥 ∈ ran
ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “
𝑍)) |
37 | 10, 36 | eqsstrid 3949 |
. 2
⊢ (𝑀 ∈ ℤ → (ran
ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “
𝑍)) |
38 | 7 | uztrn2 12457 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍) |
39 | 38 | ex 416 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍)) |
40 | 39 | ssrdv 3907 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍) |
41 | 40 | adantl 485 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍) |
42 | | df-ss 3883 |
. . . . . 6
⊢
((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥)) |
43 | 41, 42 | sylib 221 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) →
((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥)) |
44 | 18 | sseli 3896 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ) |
45 | 44 | adantl 485 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ) |
46 | | fnfvelrn 6901 |
. . . . . . 7
⊢
((ℤ≥ Fn ℤ ∧ 𝑥 ∈ ℤ) →
(ℤ≥‘𝑥) ∈ ran
ℤ≥) |
47 | 16, 45, 46 | sylancr 590 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ ran
ℤ≥) |
48 | | elrestr 16933 |
. . . . . 6
⊢ ((ran
ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧
(ℤ≥‘𝑥) ∈ ran ℤ≥) →
((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥
↾t 𝑍)) |
49 | 6, 8, 47, 48 | mp3an12i 1467 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) →
((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥
↾t 𝑍)) |
50 | 43, 49 | eqeltrrd 2839 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran
ℤ≥ ↾t 𝑍)) |
51 | 50 | ralrimiva 3105 |
. . 3
⊢ (𝑀 ∈ ℤ →
∀𝑥 ∈ 𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍)) |
52 | | ffun 6548 |
. . . . 5
⊢
(ℤ≥:ℤ⟶𝒫 ℤ → Fun
ℤ≥) |
53 | 3, 52 | ax-mp 5 |
. . . 4
⊢ Fun
ℤ≥ |
54 | 3 | fdmi 6557 |
. . . . 5
⊢ dom
ℤ≥ = ℤ |
55 | 18, 54 | sseqtrri 3938 |
. . . 4
⊢ 𝑍 ⊆ dom
ℤ≥ |
56 | | funimass4 6777 |
. . . 4
⊢ ((Fun
ℤ≥ ∧ 𝑍 ⊆ dom ℤ≥) →
((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)
↔ ∀𝑥 ∈
𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍))) |
57 | 53, 55, 56 | mp2an 692 |
. . 3
⊢
((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)
↔ ∀𝑥 ∈
𝑍
(ℤ≥‘𝑥) ∈ (ran ℤ≥
↾t 𝑍)) |
58 | 51, 57 | sylibr 237 |
. 2
⊢ (𝑀 ∈ ℤ →
(ℤ≥ “ 𝑍) ⊆ (ran ℤ≥
↾t 𝑍)) |
59 | 37, 58 | eqssd 3918 |
1
⊢ (𝑀 ∈ ℤ → (ran
ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍)) |