Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐴 ) ) |
2 |
1
|
sseq2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐴 ) ) ) |
3 |
2
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐴 ) ) ) ) |
4 |
|
fveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑘 ) ) |
5 |
4
|
sseq2d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑥 = 𝑘 → ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
8 |
7
|
sseq2d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐵 ) ) |
11 |
10
|
sseq2d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐵 ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝐵 → ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐵 ) ) ) ) |
13 |
|
ssid |
⊢ ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐴 ) |
14 |
13
|
2a1i |
⊢ ( 𝐴 ∈ ℤ → ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐴 ) ) ) |
15 |
|
eluznn |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) ) → 𝑘 ∈ ℕ ) |
16 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑘 ) ) |
17 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
18 |
16 17
|
sseq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ↔ ( 𝐹 ‘ 𝑘 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
19 |
18
|
rspccva |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
20 |
15 19
|
sylan2 |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) ) ) → ( 𝐹 ‘ 𝑘 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
21 |
20
|
anassrs |
⊢ ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) ) → ( 𝐹 ‘ 𝑘 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
22 |
|
sstr2 |
⊢ ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) → ( ( 𝐹 ‘ 𝑘 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
23 |
21 22
|
syl5com |
⊢ ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) ) → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
24 |
23
|
expcom |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
25 |
24
|
a2d |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝑘 ) ) → ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
26 |
3 6 9 12 14 25
|
uzind4 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐵 ) ) ) |
27 |
26
|
com12 |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐵 ) ) ) |
28 |
27
|
impr |
⊢ ( ( ∀ 𝑛 ∈ ℕ ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ) → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ 𝐵 ) ) |