Step |
Hyp |
Ref |
Expression |
0 |
|
clo1 |
⊢ ≤𝑂(1) |
1 |
|
vf |
⊢ 𝑓 |
2 |
|
cr |
⊢ ℝ |
3 |
|
cpm |
⊢ ↑pm |
4 |
2 2 3
|
co |
⊢ ( ℝ ↑pm ℝ ) |
5 |
|
vx |
⊢ 𝑥 |
6 |
|
vm |
⊢ 𝑚 |
7 |
|
vy |
⊢ 𝑦 |
8 |
1
|
cv |
⊢ 𝑓 |
9 |
8
|
cdm |
⊢ dom 𝑓 |
10 |
5
|
cv |
⊢ 𝑥 |
11 |
|
cico |
⊢ [,) |
12 |
|
cpnf |
⊢ +∞ |
13 |
10 12 11
|
co |
⊢ ( 𝑥 [,) +∞ ) |
14 |
9 13
|
cin |
⊢ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) |
15 |
7
|
cv |
⊢ 𝑦 |
16 |
15 8
|
cfv |
⊢ ( 𝑓 ‘ 𝑦 ) |
17 |
|
cle |
⊢ ≤ |
18 |
6
|
cv |
⊢ 𝑚 |
19 |
16 18 17
|
wbr |
⊢ ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 |
20 |
19 7 14
|
wral |
⊢ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 |
21 |
20 6 2
|
wrex |
⊢ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 |
22 |
21 5 2
|
wrex |
⊢ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 |
23 |
22 1 4
|
crab |
⊢ { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 } |
24 |
0 23
|
wceq |
⊢ ≤𝑂(1) = { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 } |