Step | Hyp | Ref
| Expression |
1 | | lo1f 15079 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
𝐹:dom 𝐹⟶ℝ) |
2 | | lo1bdd 15081 |
. . . 4
⊢ ((𝐹 ∈ ≤𝑂(1) ∧
𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)) |
3 | 1, 2 | mpdan 687 |
. . 3
⊢ (𝐹 ∈ ≤𝑂(1) →
∃𝑥 ∈ ℝ
∃𝑚 ∈ ℝ
∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)) |
4 | | inss1 4143 |
. . . . . . 7
⊢ (dom
𝐹 ∩ 𝐴) ⊆ dom 𝐹 |
5 | | ssralv 3967 |
. . . . . . 7
⊢ ((dom
𝐹 ∩ 𝐴) ⊆ dom 𝐹 → (∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))) |
6 | 4, 5 | ax-mp 5 |
. . . . . 6
⊢
(∀𝑦 ∈
dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)) |
7 | | elinel2 4110 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → 𝑦 ∈ 𝐴) |
8 | 7 | fvresd 6737 |
. . . . . . . . 9
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦)) |
9 | 8 | breq1d 5063 |
. . . . . . . 8
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚)) |
10 | 9 | imbi2d 344 |
. . . . . . 7
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))) |
11 | 10 | ralbiia 3087 |
. . . . . 6
⊢
(∀𝑦 ∈
(dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)) |
12 | 6, 11 | sylibr 237 |
. . . . 5
⊢
(∀𝑦 ∈
dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)) |
13 | 12 | reximi 3166 |
. . . 4
⊢
(∃𝑚 ∈
ℝ ∀𝑦 ∈
dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)) |
14 | 13 | reximi 3166 |
. . 3
⊢
(∃𝑥 ∈
ℝ ∃𝑚 ∈
ℝ ∀𝑦 ∈
dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)) |
15 | 3, 14 | syl 17 |
. 2
⊢ (𝐹 ∈ ≤𝑂(1) →
∃𝑥 ∈ ℝ
∃𝑚 ∈ ℝ
∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)) |
16 | | fssres 6585 |
. . . . 5
⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
17 | 1, 4, 16 | sylancl 589 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
18 | | resres 5864 |
. . . . . 6
⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) |
19 | | ffn 6545 |
. . . . . . . 8
⊢ (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹) |
20 | | fnresdm 6496 |
. . . . . . . 8
⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
21 | 1, 19, 20 | 3syl 18 |
. . . . . . 7
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ dom 𝐹) = 𝐹) |
22 | 21 | reseq1d 5850 |
. . . . . 6
⊢ (𝐹 ∈ ≤𝑂(1) →
((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴)) |
23 | 18, 22 | eqtr3id 2792 |
. . . . 5
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴)) |
24 | 23 | feq1d 6530 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ ↔ (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ)) |
25 | 17, 24 | mpbid 235 |
. . 3
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
26 | | lo1dm 15080 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
dom 𝐹 ⊆
ℝ) |
27 | 4, 26 | sstrid 3912 |
. . 3
⊢ (𝐹 ∈ ≤𝑂(1) →
(dom 𝐹 ∩ 𝐴) ⊆
ℝ) |
28 | | ello12 15077 |
. . 3
⊢ (((𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ ℝ) → ((𝐹 ↾ 𝐴) ∈ ≤𝑂(1) ↔
∃𝑥 ∈ ℝ
∃𝑚 ∈ ℝ
∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))) |
29 | 25, 27, 28 | syl2anc 587 |
. 2
⊢ (𝐹 ∈ ≤𝑂(1) →
((𝐹 ↾ 𝐴) ∈ ≤𝑂(1) ↔
∃𝑥 ∈ ℝ
∃𝑚 ∈ ℝ
∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))) |
30 | 15, 29 | mpbird 260 |
1
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ 𝐴) ∈
≤𝑂(1)) |