Step |
Hyp |
Ref |
Expression |
1 |
|
lo1f |
⊢ ( 𝐹 ∈ ≤𝑂(1) → 𝐹 : dom 𝐹 ⟶ ℝ ) |
2 |
|
lo1bdd |
⊢ ( ( 𝐹 ∈ ≤𝑂(1) ∧ 𝐹 : dom 𝐹 ⟶ ℝ ) → ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ dom 𝐹 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) |
3 |
1 2
|
mpdan |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ dom 𝐹 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) |
4 |
|
inss1 |
⊢ ( dom 𝐹 ∩ 𝐴 ) ⊆ dom 𝐹 |
5 |
|
ssralv |
⊢ ( ( 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 |
⊢ ( ∀ 𝑦 ∈ dom 𝐹 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) → ∀ 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) |
7 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) → 𝑦 ∈ 𝐴 ) |
8 |
7
|
fvresd |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
9 |
8
|
breq1d |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) → ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) ≤ 𝑚 ↔ ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) → ( ( 𝑥 ≤ 𝑦 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) ≤ 𝑚 ) ↔ ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) ) |
11 |
10
|
ralbiia |
⊢ ( ∀ 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) ( 𝑥 ≤ 𝑦 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) ≤ 𝑚 ) ↔ ∀ 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) ) |
12 |
6 11
|
sylibr |
⊢ ( ∀ 𝑦 ∈ dom 𝐹 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) → ∀ 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) ( 𝑥 ≤ 𝑦 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) ≤ 𝑚 ) ) |
13 |
12
|
reximi |
⊢ ( ∃ 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 |
⊢ ( ∃ 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 |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ 𝐴 ) ( 𝑥 ≤ 𝑦 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑦 ) ≤ 𝑚 ) ) |
16 |
|
fssres |
⊢ ( ( 𝐹 : dom 𝐹 ⟶ ℝ ∧ ( dom 𝐹 ∩ 𝐴 ) ⊆ dom 𝐹 ) → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐴 ) ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℝ ) |
17 |
1 4 16
|
sylancl |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐴 ) ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℝ ) |
18 |
|
resres |
⊢ ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐴 ) = ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐴 ) ) |
19 |
|
ffn |
⊢ ( 𝐹 : dom 𝐹 ⟶ ℝ → 𝐹 Fn dom 𝐹 ) |
20 |
|
fnresdm |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
21 |
1 19 20
|
3syl |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
22 |
21
|
reseq1d |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐴 ) = ( 𝐹 ↾ 𝐴 ) ) |
23 |
18 22
|
eqtr3id |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐴 ) ) = ( 𝐹 ↾ 𝐴 ) ) |
24 |
23
|
feq1d |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐴 ) ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℝ ↔ ( 𝐹 ↾ 𝐴 ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℝ ) ) |
25 |
17 24
|
mpbid |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ 𝐴 ) : ( dom 𝐹 ∩ 𝐴 ) ⟶ ℝ ) |
26 |
|
lo1dm |
⊢ ( 𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ ) |
27 |
4 26
|
sstrid |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( dom 𝐹 ∩ 𝐴 ) ⊆ ℝ ) |
28 |
|
ello12 |
⊢ ( ( ( 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 |
⊢ ( 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 |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ 𝐴 ) ∈ ≤𝑂(1) ) |