Step |
Hyp |
Ref |
Expression |
1 |
|
methaus.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
2 |
1
|
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
3 |
1
|
mopnuni |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
4 |
3
|
eleq2d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽 ) ) |
5 |
4
|
biimpar |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ ∪ 𝐽 ) → 𝑥 ∈ 𝑋 ) |
6 |
|
simpll |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
7 |
|
simplr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → 𝑥 ∈ 𝑋 ) |
8 |
|
nnrp |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ+ ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℝ+ ) |
10 |
9
|
rpreccld |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ+ ) |
11 |
10
|
rpxrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℝ* ) |
12 |
1
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 1 / 𝑛 ) ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ 𝐽 ) |
13 |
6 7 11 12
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ 𝐽 ) |
14 |
13
|
fmpttd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) : ℕ ⟶ 𝐽 ) |
15 |
14
|
frnd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ⊆ 𝐽 ) |
16 |
|
nnex |
⊢ ℕ ∈ V |
17 |
16
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ∈ V |
18 |
17
|
rnex |
⊢ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ∈ V |
19 |
18
|
elpw |
⊢ ( ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ∈ 𝒫 𝐽 ↔ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ⊆ 𝐽 ) |
20 |
15 19
|
sylibr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ∈ 𝒫 𝐽 ) |
21 |
|
omelon |
⊢ ω ∈ On |
22 |
|
nnenom |
⊢ ℕ ≈ ω |
23 |
22
|
ensymi |
⊢ ω ≈ ℕ |
24 |
|
isnumi |
⊢ ( ( ω ∈ On ∧ ω ≈ ℕ ) → ℕ ∈ dom card ) |
25 |
21 23 24
|
mp2an |
⊢ ℕ ∈ dom card |
26 |
|
ovex |
⊢ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ V |
27 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
28 |
26 27
|
fnmpti |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) Fn ℕ |
29 |
|
dffn4 |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) Fn ℕ ↔ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) : ℕ –onto→ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
30 |
28 29
|
mpbi |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) : ℕ –onto→ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
31 |
|
fodomnum |
⊢ ( ℕ ∈ dom card → ( ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) : ℕ –onto→ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ℕ ) ) |
32 |
25 30 31
|
mp2 |
⊢ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ℕ |
33 |
|
domentr |
⊢ ( ( ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ℕ ∧ ℕ ≈ ω ) → ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω ) |
34 |
32 22 33
|
mp2an |
⊢ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω |
35 |
34
|
a1i |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω ) |
36 |
|
simpll |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
37 |
|
simprl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → 𝑧 ∈ 𝐽 ) |
38 |
|
simprr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → 𝑥 ∈ 𝑧 ) |
39 |
1
|
mopni2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) |
40 |
36 37 38 39
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) |
41 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
42 |
|
simp-4r |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑥 ∈ 𝑋 ) |
43 |
|
simprl |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑦 ∈ ℕ ) |
44 |
43
|
nnrpd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑦 ∈ ℝ+ ) |
45 |
44
|
rpreccld |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 1 / 𝑦 ) ∈ ℝ+ ) |
46 |
|
blcntr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 1 / 𝑦 ) ∈ ℝ+ ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) |
47 |
41 42 45 46
|
syl3anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) |
48 |
45
|
rpxrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 1 / 𝑦 ) ∈ ℝ* ) |
49 |
|
simplrl |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑟 ∈ ℝ+ ) |
50 |
49
|
rpxrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑟 ∈ ℝ* ) |
51 |
|
nnrecre |
⊢ ( 𝑦 ∈ ℕ → ( 1 / 𝑦 ) ∈ ℝ ) |
52 |
51
|
ad2antrl |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 1 / 𝑦 ) ∈ ℝ ) |
53 |
49
|
rpred |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → 𝑟 ∈ ℝ ) |
54 |
|
simprr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 1 / 𝑦 ) < 𝑟 ) |
55 |
52 53 54
|
ltled |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 1 / 𝑦 ) ≤ 𝑟 ) |
56 |
|
ssbl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( ( 1 / 𝑦 ) ∈ ℝ* ∧ 𝑟 ∈ ℝ* ) ∧ ( 1 / 𝑦 ) ≤ 𝑟 ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
57 |
41 42 48 50 55 56
|
syl221anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
58 |
|
simplrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) |
59 |
57 58
|
sstrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) |
60 |
47 59
|
jca |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) ∧ ( 𝑦 ∈ ℕ ∧ ( 1 / 𝑦 ) < 𝑟 ) ) → ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) |
61 |
|
elrp |
⊢ ( 𝑟 ∈ ℝ+ ↔ ( 𝑟 ∈ ℝ ∧ 0 < 𝑟 ) ) |
62 |
|
nnrecl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 0 < 𝑟 ) → ∃ 𝑦 ∈ ℕ ( 1 / 𝑦 ) < 𝑟 ) |
63 |
61 62
|
sylbi |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑦 ∈ ℕ ( 1 / 𝑦 ) < 𝑟 ) |
64 |
63
|
ad2antrl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → ∃ 𝑦 ∈ ℕ ( 1 / 𝑦 ) < 𝑟 ) |
65 |
60 64
|
reximddv |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → ∃ 𝑦 ∈ ℕ ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) |
66 |
40 65
|
rexlimddv |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → ∃ 𝑦 ∈ ℕ ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) |
67 |
|
ovexd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ 𝑦 ∈ ℕ ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∈ V ) |
68 |
|
vex |
⊢ 𝑤 ∈ V |
69 |
|
oveq2 |
⊢ ( 𝑛 = 𝑦 → ( 1 / 𝑛 ) = ( 1 / 𝑦 ) ) |
70 |
69
|
oveq2d |
⊢ ( 𝑛 = 𝑦 → ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) |
71 |
70
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) = ( 𝑦 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) |
72 |
71
|
elrnmpt |
⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ↔ ∃ 𝑦 ∈ ℕ 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) ) |
73 |
68 72
|
mp1i |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → ( 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ↔ ∃ 𝑦 ∈ ℕ 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) ) |
74 |
|
eleq2 |
⊢ ( 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) → ( 𝑥 ∈ 𝑤 ↔ 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) ) |
75 |
|
sseq1 |
⊢ ( 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) → ( 𝑤 ⊆ 𝑧 ↔ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) |
76 |
74 75
|
anbi12d |
⊢ ( 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) → ( ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ↔ ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) ) |
77 |
76
|
adantl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) ∧ 𝑤 = ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ) → ( ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ↔ ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) ) |
78 |
67 73 77
|
rexxfr2d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → ( ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ↔ ∃ 𝑦 ∈ ℕ ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑦 ) ) ⊆ 𝑧 ) ) ) |
79 |
66 78
|
mpbird |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧 ) ) → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) |
80 |
79
|
expr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) |
81 |
80
|
ralrimiva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) |
82 |
|
breq1 |
⊢ ( 𝑦 = ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ( 𝑦 ≼ ω ↔ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω ) ) |
83 |
|
rexeq |
⊢ ( 𝑦 = ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ( ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ↔ ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) |
84 |
83
|
imbi2d |
⊢ ( 𝑦 = ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ( ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ↔ ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
85 |
84
|
ralbidv |
⊢ ( 𝑦 = ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ( ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
86 |
82 85
|
anbi12d |
⊢ ( 𝑦 = ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ( ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ↔ ( ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) ) |
87 |
86
|
rspcev |
⊢ ( ( ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ∈ 𝒫 𝐽 ∧ ( ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ ran ( 𝑛 ∈ ℕ ↦ ( 𝑥 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) → ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
88 |
20 35 81 87
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
89 |
5 88
|
syldan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ ∪ 𝐽 ) → ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
90 |
89
|
ralrimiva |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∀ 𝑥 ∈ ∪ 𝐽 ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) |
91 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
92 |
91
|
is1stc2 |
⊢ ( 𝐽 ∈ 1stω ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∃ 𝑦 ∈ 𝒫 𝐽 ( 𝑦 ≼ ω ∧ ∀ 𝑧 ∈ 𝐽 ( 𝑥 ∈ 𝑧 → ∃ 𝑤 ∈ 𝑦 ( 𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧 ) ) ) ) ) |
93 |
2 90 92
|
sylanbrc |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ 1stω ) |