| Step | Hyp | Ref
| Expression |
| 1 | | lo1f 15554 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
𝐹:dom 𝐹⟶ℝ) |
| 2 | | lo1bdd 15556 |
. . . 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 4237 |
. . . . . . 7
⊢ (dom
𝐹 ∩ 𝐴) ⊆ dom 𝐹 |
| 5 | | ssralv 4052 |
. . . . . . 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 4202 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → 𝑦 ∈ 𝐴) |
| 8 | 7 | fvresd 6926 |
. . . . . . . . 9
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦)) |
| 9 | 8 | breq1d 5153 |
. . . . . . . 8
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚)) |
| 10 | 9 | imbi2d 340 |
. . . . . . 7
⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))) |
| 11 | 10 | ralbiia 3091 |
. . . . . 6
⊢
(∀𝑦 ∈
(dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)) |
| 12 | 6, 11 | sylibr 234 |
. . . . 5
⊢
(∀𝑦 ∈
dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)) |
| 13 | 12 | reximi 3084 |
. . . 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 3084 |
. . 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 6774 |
. . . . 5
⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
| 17 | 1, 4, 16 | sylancl 586 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
| 18 | | resres 6010 |
. . . . . 6
⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) |
| 19 | | ffn 6736 |
. . . . . . . 8
⊢ (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹) |
| 20 | | fnresdm 6687 |
. . . . . . . 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 5996 |
. . . . . 6
⊢ (𝐹 ∈ ≤𝑂(1) →
((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴)) |
| 23 | 18, 22 | eqtr3id 2791 |
. . . . 5
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴)) |
| 24 | 23 | feq1d 6720 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ ↔ (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ)) |
| 25 | 17, 24 | mpbid 232 |
. . 3
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ) |
| 26 | | lo1dm 15555 |
. . . 4
⊢ (𝐹 ∈ ≤𝑂(1) →
dom 𝐹 ⊆
ℝ) |
| 27 | 4, 26 | sstrid 3995 |
. . 3
⊢ (𝐹 ∈ ≤𝑂(1) →
(dom 𝐹 ∩ 𝐴) ⊆
ℝ) |
| 28 | | ello12 15552 |
. . 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 584 |
. 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 257 |
1
⊢ (𝐹 ∈ ≤𝑂(1) →
(𝐹 ↾ 𝐴) ∈
≤𝑂(1)) |