Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
⊢ 𝐹 = ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
2 |
1
|
metustss |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → 𝐴 ⊆ ( 𝑋 × 𝑋 ) ) |
3 |
|
cnvss |
⊢ ( 𝐴 ⊆ ( 𝑋 × 𝑋 ) → ◡ 𝐴 ⊆ ◡ ( 𝑋 × 𝑋 ) ) |
4 |
|
cnvxp |
⊢ ◡ ( 𝑋 × 𝑋 ) = ( 𝑋 × 𝑋 ) |
5 |
3 4
|
sseqtrdi |
⊢ ( 𝐴 ⊆ ( 𝑋 × 𝑋 ) → ◡ 𝐴 ⊆ ( 𝑋 × 𝑋 ) ) |
6 |
2 5
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → ◡ 𝐴 ⊆ ( 𝑋 × 𝑋 ) ) |
7 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
8 |
|
simpr1r |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) → 𝑞 ∈ 𝑋 ) |
9 |
8
|
3anassrs |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝑞 ∈ 𝑋 ) |
10 |
|
simpr1l |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) → 𝑝 ∈ 𝑋 ) |
11 |
10
|
3anassrs |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝑝 ∈ 𝑋 ) |
12 |
|
psmetsym |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋 ) → ( 𝑞 𝐷 𝑝 ) = ( 𝑝 𝐷 𝑞 ) ) |
13 |
7 9 11 12
|
syl3anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝑞 𝐷 𝑝 ) = ( 𝑝 𝐷 𝑞 ) ) |
14 |
|
df-ov |
⊢ ( 𝑞 𝐷 𝑝 ) = ( 𝐷 ‘ 〈 𝑞 , 𝑝 〉 ) |
15 |
|
df-ov |
⊢ ( 𝑝 𝐷 𝑞 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) |
16 |
13 14 15
|
3eqtr3g |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝐷 ‘ 〈 𝑞 , 𝑝 〉 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ) |
17 |
16
|
eleq1d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( 𝐷 ‘ 〈 𝑞 , 𝑝 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ) ) |
18 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
19 |
|
ffun |
⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → Fun 𝐷 ) |
20 |
7 18 19
|
3syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → Fun 𝐷 ) |
21 |
|
simpllr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
22 |
21
|
ancomd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋 ) ) |
23 |
|
opelxpi |
⊢ ( ( 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋 ) → 〈 𝑞 , 𝑝 〉 ∈ ( 𝑋 × 𝑋 ) ) |
24 |
22 23
|
syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑞 , 𝑝 〉 ∈ ( 𝑋 × 𝑋 ) ) |
25 |
|
fdm |
⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
26 |
7 18 25
|
3syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
27 |
24 26
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑞 , 𝑝 〉 ∈ dom 𝐷 ) |
28 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑞 , 𝑝 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑞 , 𝑝 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑞 , 𝑝 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
29 |
20 27 28
|
syl2anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( 𝐷 ‘ 〈 𝑞 , 𝑝 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑞 , 𝑝 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
30 |
|
opelxpi |
⊢ ( ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) → 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
31 |
21 30
|
syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
32 |
31 26
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) |
33 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
34 |
20 32 33
|
syl2anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
35 |
17 29 34
|
3bitr3d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 〈 𝑞 , 𝑝 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
36 |
|
simpr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
37 |
36
|
eleq2d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 〈 𝑞 , 𝑝 〉 ∈ 𝐴 ↔ 〈 𝑞 , 𝑝 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
38 |
36
|
eleq2d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
39 |
35 37 38
|
3bitr4d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 〈 𝑞 , 𝑝 〉 ∈ 𝐴 ↔ 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) ) |
40 |
|
eqid |
⊢ ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
41 |
40
|
elrnmpt |
⊢ ( 𝐴 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝐴 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
42 |
41
|
ibi |
⊢ ( 𝐴 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
43 |
42 1
|
eleq2s |
⊢ ( 𝐴 ∈ 𝐹 → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
44 |
43
|
ad2antlr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
45 |
39 44
|
r19.29a |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → ( 〈 𝑞 , 𝑝 〉 ∈ 𝐴 ↔ 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) ) |
46 |
|
df-br |
⊢ ( 𝑝 ◡ 𝐴 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ ◡ 𝐴 ) |
47 |
|
vex |
⊢ 𝑝 ∈ V |
48 |
|
vex |
⊢ 𝑞 ∈ V |
49 |
47 48
|
opelcnv |
⊢ ( 〈 𝑝 , 𝑞 〉 ∈ ◡ 𝐴 ↔ 〈 𝑞 , 𝑝 〉 ∈ 𝐴 ) |
50 |
46 49
|
bitri |
⊢ ( 𝑝 ◡ 𝐴 𝑞 ↔ 〈 𝑞 , 𝑝 〉 ∈ 𝐴 ) |
51 |
|
df-br |
⊢ ( 𝑝 𝐴 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
52 |
45 50 51
|
3bitr4g |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → ( 𝑝 ◡ 𝐴 𝑞 ↔ 𝑝 𝐴 𝑞 ) ) |
53 |
52
|
3impb |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) → ( 𝑝 ◡ 𝐴 𝑞 ↔ 𝑝 𝐴 𝑞 ) ) |
54 |
6 2 53
|
eqbrrdva |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → ◡ 𝐴 = 𝐴 ) |