Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
⊢ 𝐹 = ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
2 |
1
|
metust |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) ) |
3 |
|
cfilufbas |
⊢ ( ( ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) → 𝐶 ∈ ( fBas ‘ 𝑋 ) ) |
4 |
2 3
|
sylan |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) → 𝐶 ∈ ( fBas ‘ 𝑋 ) ) |
5 |
|
simpllr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
6 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
7 |
|
ffun |
⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → Fun 𝐷 ) |
8 |
5 6 7
|
3syl |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → Fun 𝐷 ) |
9 |
2
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) ) |
10 |
|
simplr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) |
11 |
1
|
metustfbas |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) ) |
13 |
|
cnvimass |
⊢ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ⊆ dom 𝐷 |
14 |
|
fdm |
⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
15 |
5 6 14
|
3syl |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
16 |
13 15
|
sseqtrid |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
17 |
|
simpr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
18 |
17
|
rphalfcld |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( 𝑥 / 2 ) ∈ ℝ+ ) |
19 |
|
eqidd |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ) |
20 |
|
oveq2 |
⊢ ( 𝑎 = ( 𝑥 / 2 ) → ( 0 [,) 𝑎 ) = ( 0 [,) ( 𝑥 / 2 ) ) ) |
21 |
20
|
imaeq2d |
⊢ ( 𝑎 = ( 𝑥 / 2 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ) |
22 |
21
|
rspceeqv |
⊢ ( ( ( 𝑥 / 2 ) ∈ ℝ+ ∧ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
23 |
18 19 22
|
syl2anc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
24 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
25 |
24
|
biimpar |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ∈ 𝐹 ) |
26 |
5 23 25
|
syl2anc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ∈ 𝐹 ) |
27 |
|
0xr |
⊢ 0 ∈ ℝ* |
28 |
27
|
a1i |
⊢ ( 𝑥 ∈ ℝ+ → 0 ∈ ℝ* ) |
29 |
|
rpxr |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ* ) |
30 |
|
0le0 |
⊢ 0 ≤ 0 |
31 |
30
|
a1i |
⊢ ( 𝑥 ∈ ℝ+ → 0 ≤ 0 ) |
32 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
33 |
32
|
rehalfcld |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ∈ ℝ ) |
34 |
|
rphalflt |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) < 𝑥 ) |
35 |
33 32 34
|
ltled |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 / 2 ) ≤ 𝑥 ) |
36 |
|
icossico |
⊢ ( ( ( 0 ∈ ℝ* ∧ 𝑥 ∈ ℝ* ) ∧ ( 0 ≤ 0 ∧ ( 𝑥 / 2 ) ≤ 𝑥 ) ) → ( 0 [,) ( 𝑥 / 2 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
37 |
28 29 31 35 36
|
syl22anc |
⊢ ( 𝑥 ∈ ℝ+ → ( 0 [,) ( 𝑥 / 2 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
38 |
|
imass2 |
⊢ ( ( 0 [,) ( 𝑥 / 2 ) ) ⊆ ( 0 [,) 𝑥 ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
39 |
17 37 38
|
3syl |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
40 |
|
sseq1 |
⊢ ( 𝑤 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) → ( 𝑤 ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ↔ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) ) |
41 |
40
|
rspcev |
⊢ ( ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ∈ 𝐹 ∧ ( ◡ 𝐷 “ ( 0 [,) ( 𝑥 / 2 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) → ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
42 |
26 39 41
|
syl2anc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
43 |
|
elfg |
⊢ ( 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) → ( ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ↔ ( ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) ) ) |
44 |
43
|
biimpar |
⊢ ( ( 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) ∧ ( ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) ) → ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) |
45 |
12 16 42 44
|
syl12anc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) |
46 |
|
cfiluexsm |
⊢ ( ( ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
47 |
9 10 45 46
|
syl3anc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) |
48 |
|
funimass2 |
⊢ ( ( Fun 𝐷 ∧ ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) ) → ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
49 |
48
|
ex |
⊢ ( Fun 𝐷 → ( ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) → ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
50 |
49
|
reximdv |
⊢ ( Fun 𝐷 → ( ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑥 ) ) → ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
51 |
8 47 50
|
sylc |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
52 |
51
|
ralrimiva |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
53 |
4 52
|
jca |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) → ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
54 |
|
simprl |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) → 𝐶 ∈ ( fBas ‘ 𝑋 ) ) |
55 |
|
oveq2 |
⊢ ( 𝑥 = 𝑎 → ( 0 [,) 𝑥 ) = ( 0 [,) 𝑎 ) ) |
56 |
55
|
sseq2d |
⊢ ( 𝑥 = 𝑎 → ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ) ) |
57 |
56
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ) ) |
58 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
59 |
58
|
simprd |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
60 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → 𝑎 ∈ ℝ+ ) |
61 |
57 59 60
|
rspcdva |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ) |
62 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
63 |
|
nfv |
⊢ Ⅎ 𝑦 𝐶 ∈ ( fBas ‘ 𝑋 ) |
64 |
|
nfcv |
⊢ Ⅎ 𝑦 ℝ+ |
65 |
|
nfre1 |
⊢ Ⅎ 𝑦 ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) |
66 |
64 65
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) |
67 |
63 66
|
nfan |
⊢ Ⅎ 𝑦 ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) |
68 |
62 67
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
69 |
|
nfv |
⊢ Ⅎ 𝑦 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) |
70 |
68 69
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) |
71 |
|
nfv |
⊢ Ⅎ 𝑦 𝑎 ∈ ℝ+ |
72 |
70 71
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) |
73 |
|
nfv |
⊢ Ⅎ 𝑦 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 |
74 |
72 73
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
75 |
54
|
ad4antr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → 𝐶 ∈ ( fBas ‘ 𝑋 ) ) |
76 |
|
fbelss |
⊢ ( ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐶 ) → 𝑦 ⊆ 𝑋 ) |
77 |
75 76
|
sylancom |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → 𝑦 ⊆ 𝑋 ) |
78 |
|
xpss12 |
⊢ ( ( 𝑦 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑋 ) → ( 𝑦 × 𝑦 ) ⊆ ( 𝑋 × 𝑋 ) ) |
79 |
77 77 78
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → ( 𝑦 × 𝑦 ) ⊆ ( 𝑋 × 𝑋 ) ) |
80 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
81 |
80 6 14
|
3syl |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
82 |
79 81
|
sseqtrrd |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ∧ 𝑦 ∈ 𝐶 ) → ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) |
83 |
82
|
ex |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ( 𝑦 ∈ 𝐶 → ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) ) |
84 |
74 83
|
ralrimi |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∀ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) |
85 |
|
r19.29r |
⊢ ( ( ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ∃ 𝑦 ∈ 𝐶 ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) ) |
86 |
|
sseqin2 |
⊢ ( ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ↔ ( dom 𝐷 ∩ ( 𝑦 × 𝑦 ) ) = ( 𝑦 × 𝑦 ) ) |
87 |
86
|
biimpi |
⊢ ( ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 → ( dom 𝐷 ∩ ( 𝑦 × 𝑦 ) ) = ( 𝑦 × 𝑦 ) ) |
88 |
87
|
adantl |
⊢ ( ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ( dom 𝐷 ∩ ( 𝑦 × 𝑦 ) ) = ( 𝑦 × 𝑦 ) ) |
89 |
|
dminss |
⊢ ( dom 𝐷 ∩ ( 𝑦 × 𝑦 ) ) ⊆ ( ◡ 𝐷 “ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ) |
90 |
88 89
|
eqsstrrdi |
⊢ ( ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ) ) |
91 |
|
imass2 |
⊢ ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) → ( ◡ 𝐷 “ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
92 |
91
|
adantr |
⊢ ( ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ( ◡ 𝐷 “ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
93 |
90 92
|
sstrd |
⊢ ( ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
94 |
93
|
reximi |
⊢ ( ∃ 𝑦 ∈ 𝐶 ( ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
95 |
85 94
|
syl |
⊢ ( ( ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑎 ) ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ dom 𝐷 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
96 |
61 84 95
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
97 |
|
r19.41v |
⊢ ( ∃ 𝑦 ∈ 𝐶 ( ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ↔ ( ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ) |
98 |
|
sstr |
⊢ ( ( ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
99 |
98
|
reximi |
⊢ ( ∃ 𝑦 ∈ 𝐶 ( ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
100 |
97 99
|
sylbir |
⊢ ( ( ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
101 |
96 100
|
sylancom |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
102 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑤 ∈ 𝐹 ) ∧ 𝑤 ⊆ 𝑣 ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
103 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑤 ∈ 𝐹 ) ∧ 𝑤 ⊆ 𝑣 ) → 𝑤 ∈ 𝐹 ) |
104 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝑤 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
105 |
104
|
biimpa |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑤 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
106 |
102 103 105
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑤 ∈ 𝐹 ) ∧ 𝑤 ⊆ 𝑣 ) → ∃ 𝑎 ∈ ℝ+ 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
107 |
|
r19.41v |
⊢ ( ∃ 𝑎 ∈ ℝ+ ( 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝑤 ⊆ 𝑣 ) ↔ ( ∃ 𝑎 ∈ ℝ+ 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝑤 ⊆ 𝑣 ) ) |
108 |
|
sseq1 |
⊢ ( 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) → ( 𝑤 ⊆ 𝑣 ↔ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) ) |
109 |
108
|
biimpa |
⊢ ( ( 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝑤 ⊆ 𝑣 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
110 |
109
|
reximi |
⊢ ( ∃ 𝑎 ∈ ℝ+ ( 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝑤 ⊆ 𝑣 ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
111 |
107 110
|
sylbir |
⊢ ( ( ∃ 𝑎 ∈ ℝ+ 𝑤 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝑤 ⊆ 𝑣 ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
112 |
106 111
|
sylancom |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ∧ 𝑤 ∈ 𝐹 ) ∧ 𝑤 ⊆ 𝑣 ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
113 |
11
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) ) |
114 |
|
elfg |
⊢ ( 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) → ( 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ↔ ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 ) ) ) |
115 |
114
|
biimpa |
⊢ ( ( 𝐹 ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 ) ) |
116 |
113 115
|
sylancom |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 ) ) |
117 |
116
|
simprd |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ∃ 𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 ) |
118 |
112 117
|
r19.29a |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ∃ 𝑎 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ 𝑣 ) |
119 |
101 118
|
r19.29a |
⊢ ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ∧ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) → ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
120 |
119
|
ralrimiva |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) → ∀ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) |
121 |
2
|
adantr |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) → ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) ) |
122 |
|
iscfilu |
⊢ ( ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∈ ( UnifOn ‘ 𝑋 ) → ( 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ↔ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) ) ) |
123 |
121 122
|
syl |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) → ( 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ↔ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑣 ∈ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ∃ 𝑦 ∈ 𝐶 ( 𝑦 × 𝑦 ) ⊆ 𝑣 ) ) ) |
124 |
54 120 123
|
mpbir2and |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) → 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ) |
125 |
53 124
|
impbida |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝐶 ∈ ( CauFilu ‘ ( ( 𝑋 × 𝑋 ) filGen 𝐹 ) ) ↔ ( 𝐶 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐶 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) ) |