Step |
Hyp |
Ref |
Expression |
1 |
|
metdscn.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) ) |
2 |
|
n0 |
⊢ ( 𝑆 ≠ ∅ ↔ ∃ 𝑧 𝑧 ∈ 𝑆 ) |
3 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
4 |
1
|
metdsf |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
5 |
3 4
|
sylan |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
7 |
6
|
ffnd |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 Fn 𝑋 ) |
8 |
5
|
adantr |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
9 |
|
simprr |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝑤 ∈ 𝑋 ) |
10 |
8 9
|
ffvelrnd |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ) |
11 |
|
eliccxr |
⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ) |
12 |
10 11
|
syl |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ) |
13 |
|
simpll |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
14 |
|
simpr |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ 𝑋 ) |
15 |
14
|
sselda |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝑧 ∈ 𝑋 ) |
16 |
15
|
adantrr |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 𝑧 ∈ 𝑋 ) |
17 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ) → ( 𝑧 𝐷 𝑤 ) ∈ ℝ ) |
18 |
13 16 9 17
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝑧 𝐷 𝑤 ) ∈ ℝ ) |
19 |
|
elxrge0 |
⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝑤 ) ) ) |
20 |
19
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝑤 ) ) |
21 |
10 20
|
syl |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → 0 ≤ ( 𝐹 ‘ 𝑤 ) ) |
22 |
1
|
metdsle |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) ) |
23 |
3 22
|
sylanl1 |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) ) |
24 |
|
xrrege0 |
⊢ ( ( ( ( 𝐹 ‘ 𝑤 ) ∈ ℝ* ∧ ( 𝑧 𝐷 𝑤 ) ∈ ℝ ) ∧ ( 0 ≤ ( 𝐹 ‘ 𝑤 ) ∧ ( 𝐹 ‘ 𝑤 ) ≤ ( 𝑧 𝐷 𝑤 ) ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) |
25 |
12 18 21 23 24
|
syl22anc |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) |
26 |
25
|
anassrs |
⊢ ( ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) ∧ 𝑤 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) |
27 |
26
|
ralrimiva |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → ∀ 𝑤 ∈ 𝑋 ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) |
28 |
|
ffnfv |
⊢ ( 𝐹 : 𝑋 ⟶ ℝ ↔ ( 𝐹 Fn 𝑋 ∧ ∀ 𝑤 ∈ 𝑋 ( 𝐹 ‘ 𝑤 ) ∈ ℝ ) ) |
29 |
7 27 28
|
sylanbrc |
⊢ ( ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝑧 ∈ 𝑆 ) → 𝐹 : 𝑋 ⟶ ℝ ) |
30 |
29
|
ex |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑧 ∈ 𝑆 → 𝐹 : 𝑋 ⟶ ℝ ) ) |
31 |
30
|
exlimdv |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( ∃ 𝑧 𝑧 ∈ 𝑆 → 𝐹 : 𝑋 ⟶ ℝ ) ) |
32 |
2 31
|
syl5bi |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ≠ ∅ → 𝐹 : 𝑋 ⟶ ℝ ) ) |
33 |
32
|
3impia |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ) → 𝐹 : 𝑋 ⟶ ℝ ) |