Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
⊢ 𝐹 = ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
2 |
|
relres |
⊢ Rel ( I ↾ 𝑋 ) |
3 |
2
|
a1i |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → Rel ( I ↾ 𝑋 ) ) |
4 |
|
vex |
⊢ 𝑞 ∈ V |
5 |
4
|
brresi |
⊢ ( 𝑝 ( I ↾ 𝑋 ) 𝑞 ↔ ( 𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞 ) ) |
6 |
|
df-br |
⊢ ( 𝑝 ( I ↾ 𝑋 ) 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) |
7 |
4
|
ideq |
⊢ ( 𝑝 I 𝑞 ↔ 𝑝 = 𝑞 ) |
8 |
7
|
anbi2i |
⊢ ( ( 𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞 ) ↔ ( 𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞 ) ) |
9 |
5 6 8
|
3bitr3i |
⊢ ( 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ↔ ( 𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞 ) ) |
10 |
9
|
biimpi |
⊢ ( 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) → ( 𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞 ) ) |
11 |
10
|
ad2antlr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( 𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞 ) ) |
12 |
11
|
simprd |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 𝑝 = 𝑞 ) |
13 |
|
df-ov |
⊢ ( 𝑝 𝐷 𝑝 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑝 〉 ) |
14 |
|
opeq2 |
⊢ ( 𝑝 = 𝑞 → 〈 𝑝 , 𝑝 〉 = 〈 𝑝 , 𝑞 〉 ) |
15 |
14
|
fveq2d |
⊢ ( 𝑝 = 𝑞 → ( 𝐷 ‘ 〈 𝑝 , 𝑝 〉 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ) |
16 |
13 15
|
syl5eq |
⊢ ( 𝑝 = 𝑞 → ( 𝑝 𝐷 𝑝 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ) |
17 |
12 16
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( 𝑝 𝐷 𝑝 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ) |
18 |
|
simplll |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
19 |
11
|
simpld |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 𝑝 ∈ 𝑋 ) |
20 |
|
psmet0 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( 𝑝 𝐷 𝑝 ) = 0 ) |
21 |
18 19 20
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( 𝑝 𝐷 𝑝 ) = 0 ) |
22 |
17 21
|
eqtr3d |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) = 0 ) |
23 |
|
0xr |
⊢ 0 ∈ ℝ* |
24 |
|
rpxr |
⊢ ( 𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ* ) |
25 |
|
rpgt0 |
⊢ ( 𝑎 ∈ ℝ+ → 0 < 𝑎 ) |
26 |
|
lbico1 |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 0 < 𝑎 ) → 0 ∈ ( 0 [,) 𝑎 ) ) |
27 |
23 24 25 26
|
mp3an2i |
⊢ ( 𝑎 ∈ ℝ+ → 0 ∈ ( 0 [,) 𝑎 ) ) |
28 |
27
|
adantl |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 0 ∈ ( 0 [,) 𝑎 ) ) |
29 |
22 28
|
eqeltrd |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ) |
30 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
31 |
30
|
ffund |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → Fun 𝐷 ) |
32 |
31
|
ad3antrrr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → Fun 𝐷 ) |
33 |
12 19
|
eqeltrrd |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 𝑞 ∈ 𝑋 ) |
34 |
19 33
|
opelxpd |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
35 |
30
|
fdmd |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
36 |
35
|
ad3antrrr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
37 |
34 36
|
eleqtrrd |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) |
38 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
39 |
32 37 38
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
40 |
29 39
|
mpbid |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) → 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
41 |
40
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
42 |
|
simpr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
43 |
41 42
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
44 |
|
simplr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) → 𝐴 ∈ 𝐹 ) |
45 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
47 |
44 46
|
mpbid |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
48 |
43 47
|
r19.29a |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
49 |
48
|
ex |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → ( 〈 𝑝 , 𝑞 〉 ∈ ( I ↾ 𝑋 ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) ) |
50 |
3 49
|
relssdv |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → ( I ↾ 𝑋 ) ⊆ 𝐴 ) |