| Step |
Hyp |
Ref |
Expression |
| 1 |
|
bcth.2 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
|
cmetmet |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
| 3 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 4 |
2 3
|
syl |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 5 |
1
|
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 6 |
5
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → 𝐽 ∈ Top ) |
| 7 |
|
ffvelrn |
⊢ ( ( 𝑀 : ℕ ⟶ 𝐽 ∧ 𝑘 ∈ ℕ ) → ( 𝑀 ‘ 𝑘 ) ∈ 𝐽 ) |
| 8 |
|
elssuni |
⊢ ( ( 𝑀 ‘ 𝑘 ) ∈ 𝐽 → ( 𝑀 ‘ 𝑘 ) ⊆ ∪ 𝐽 ) |
| 9 |
7 8
|
syl |
⊢ ( ( 𝑀 : ℕ ⟶ 𝐽 ∧ 𝑘 ∈ ℕ ) → ( 𝑀 ‘ 𝑘 ) ⊆ ∪ 𝐽 ) |
| 10 |
9
|
adantll |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑀 ‘ 𝑘 ) ⊆ ∪ 𝐽 ) |
| 11 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
| 12 |
11
|
clsval2 |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑀 ‘ 𝑘 ) ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) |
| 13 |
6 10 12
|
syl2anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) |
| 14 |
1
|
mopnuni |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
| 15 |
14
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → 𝑋 = ∪ 𝐽 ) |
| 16 |
13 15
|
eqeq12d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 ↔ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) = ∪ 𝐽 ) ) |
| 17 |
|
difeq2 |
⊢ ( ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) = ∪ 𝐽 → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ( ∪ 𝐽 ∖ ∪ 𝐽 ) ) |
| 18 |
|
difid |
⊢ ( ∪ 𝐽 ∖ ∪ 𝐽 ) = ∅ |
| 19 |
17 18
|
eqtrdi |
⊢ ( ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) = ∪ 𝐽 → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ∅ ) |
| 20 |
|
difss |
⊢ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ⊆ ∪ 𝐽 |
| 21 |
11
|
ntropn |
⊢ ( ( 𝐽 ∈ Top ∧ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ⊆ ∪ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ∈ 𝐽 ) |
| 22 |
6 20 21
|
sylancl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ∈ 𝐽 ) |
| 23 |
|
elssuni |
⊢ ( ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ∈ 𝐽 → ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ⊆ ∪ 𝐽 ) |
| 24 |
22 23
|
syl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ⊆ ∪ 𝐽 ) |
| 25 |
|
dfss4 |
⊢ ( ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ⊆ ∪ 𝐽 ↔ ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) |
| 26 |
24 25
|
sylib |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) |
| 27 |
|
id |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℕ ) |
| 28 |
|
elfvdm |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 ∈ dom ∞Met ) |
| 29 |
|
difexg |
⊢ ( 𝑋 ∈ dom ∞Met → ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ∈ V ) |
| 30 |
28 29
|
syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ∈ V ) |
| 31 |
30
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ∈ V ) |
| 32 |
|
fveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ 𝑘 ) ) |
| 33 |
32
|
difeq2d |
⊢ ( 𝑥 = 𝑘 → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) = ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 34 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) |
| 35 |
33 34
|
fvmptg |
⊢ ( ( 𝑘 ∈ ℕ ∧ ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ∈ V ) → ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 36 |
27 31 35
|
syl2anr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 37 |
15
|
difeq1d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) = ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 38 |
36 37
|
eqtrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 39 |
38
|
fveq2d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) |
| 40 |
26 39
|
eqtr4d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ) |
| 41 |
40
|
eqeq1d |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( ∪ 𝐽 ∖ ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) ) = ∅ ↔ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) ) |
| 42 |
19 41
|
syl5ib |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑘 ) ) ) ) = ∪ 𝐽 → ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) ) |
| 43 |
16 42
|
sylbid |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 → ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) ) |
| 44 |
43
|
ralimdva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∀ 𝑘 ∈ ℕ ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 → ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) ) |
| 45 |
4 44
|
sylan |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∀ 𝑘 ∈ ℕ ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 → ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) ) |
| 46 |
|
ffvelrn |
⊢ ( ( 𝑀 : ℕ ⟶ 𝐽 ∧ 𝑥 ∈ ℕ ) → ( 𝑀 ‘ 𝑥 ) ∈ 𝐽 ) |
| 47 |
14
|
difeq1d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) = ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑥 ) ) ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑀 ‘ 𝑥 ) ∈ 𝐽 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) = ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑥 ) ) ) |
| 49 |
11
|
opncld |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑀 ‘ 𝑥 ) ∈ 𝐽 ) → ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 50 |
5 49
|
sylan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑀 ‘ 𝑥 ) ∈ 𝐽 ) → ( ∪ 𝐽 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 51 |
48 50
|
eqeltrd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑀 ‘ 𝑥 ) ∈ 𝐽 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 52 |
46 51
|
sylan2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑀 : ℕ ⟶ 𝐽 ∧ 𝑥 ∈ ℕ ) ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 53 |
52
|
anassrs |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑥 ∈ ℕ ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 54 |
53
|
ralrimiva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∀ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 55 |
4 54
|
sylan |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∀ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
| 56 |
34
|
fmpt |
⊢ ( ∀ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ) |
| 57 |
55 56
|
sylib |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ) |
| 58 |
|
nne |
⊢ ( ¬ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ↔ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) |
| 59 |
58
|
ralbii |
⊢ ( ∀ 𝑘 ∈ ℕ ¬ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ↔ ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ) |
| 60 |
|
ralnex |
⊢ ( ∀ 𝑘 ∈ ℕ ¬ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ↔ ¬ ∃ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ) |
| 61 |
59 60
|
bitr3i |
⊢ ( ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ ↔ ¬ ∃ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ) |
| 62 |
1
|
bcth |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ∧ ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ≠ ∅ ) → ∃ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ) |
| 63 |
62
|
3expia |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ) → ( ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ≠ ∅ → ∃ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ ) ) |
| 64 |
63
|
necon1bd |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ) → ( ¬ ∃ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) ≠ ∅ → ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ ) ) |
| 65 |
61 64
|
syl5bi |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) : ℕ ⟶ ( Clsd ‘ 𝐽 ) ) → ( ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ → ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ ) ) |
| 66 |
57 65
|
syldan |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∀ 𝑘 ∈ ℕ ( ( int ‘ 𝐽 ) ‘ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) ) = ∅ → ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ ) ) |
| 67 |
|
difeq2 |
⊢ ( ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ → ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ) = ( ∪ 𝐽 ∖ ∅ ) ) |
| 68 |
|
difexg |
⊢ ( 𝑋 ∈ dom ∞Met → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ V ) |
| 69 |
28 68
|
syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ V ) |
| 70 |
69
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑥 ∈ ℕ ) → ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ V ) |
| 71 |
70
|
ralrimiva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∀ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ V ) |
| 72 |
34
|
fnmpt |
⊢ ( ∀ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ∈ V → ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) Fn ℕ ) |
| 73 |
|
fniunfv |
⊢ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) Fn ℕ → ∪ 𝑘 ∈ ℕ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) |
| 74 |
71 72 73
|
3syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ 𝑘 ∈ ℕ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) |
| 75 |
36
|
iuneq2dv |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ 𝑘 ∈ ℕ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ∪ 𝑘 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) ) |
| 76 |
33
|
cbviunv |
⊢ ∪ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) = ∪ 𝑘 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑘 ) ) |
| 77 |
75 76
|
eqtr4di |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ 𝑘 ∈ ℕ ( ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ‘ 𝑘 ) = ∪ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) |
| 78 |
74 77
|
eqtr3d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) = ∪ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) |
| 79 |
|
iundif2 |
⊢ ∪ 𝑥 ∈ ℕ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) = ( 𝑋 ∖ ∩ 𝑥 ∈ ℕ ( 𝑀 ‘ 𝑥 ) ) |
| 80 |
78 79
|
eqtrdi |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) = ( 𝑋 ∖ ∩ 𝑥 ∈ ℕ ( 𝑀 ‘ 𝑥 ) ) ) |
| 81 |
|
ffn |
⊢ ( 𝑀 : ℕ ⟶ 𝐽 → 𝑀 Fn ℕ ) |
| 82 |
81
|
adantl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → 𝑀 Fn ℕ ) |
| 83 |
|
fniinfv |
⊢ ( 𝑀 Fn ℕ → ∩ 𝑥 ∈ ℕ ( 𝑀 ‘ 𝑥 ) = ∩ ran 𝑀 ) |
| 84 |
82 83
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ 𝑥 ∈ ℕ ( 𝑀 ‘ 𝑥 ) = ∩ ran 𝑀 ) |
| 85 |
84
|
difeq2d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( 𝑋 ∖ ∩ 𝑥 ∈ ℕ ( 𝑀 ‘ 𝑥 ) ) = ( 𝑋 ∖ ∩ ran 𝑀 ) ) |
| 86 |
14
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → 𝑋 = ∪ 𝐽 ) |
| 87 |
86
|
difeq1d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( 𝑋 ∖ ∩ ran 𝑀 ) = ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) |
| 88 |
80 85 87
|
3eqtrd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) = ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) |
| 89 |
88
|
fveq2d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) ) |
| 90 |
89
|
difeq2d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ) = ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) ) ) |
| 91 |
5
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → 𝐽 ∈ Top ) |
| 92 |
|
1nn |
⊢ 1 ∈ ℕ |
| 93 |
|
biidd |
⊢ ( 𝑘 = 1 → ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) ↔ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) ) ) |
| 94 |
|
fnfvelrn |
⊢ ( ( 𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ ) → ( 𝑀 ‘ 𝑘 ) ∈ ran 𝑀 ) |
| 95 |
82 94
|
sylan |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ( 𝑀 ‘ 𝑘 ) ∈ ran 𝑀 ) |
| 96 |
|
intss1 |
⊢ ( ( 𝑀 ‘ 𝑘 ) ∈ ran 𝑀 → ∩ ran 𝑀 ⊆ ( 𝑀 ‘ 𝑘 ) ) |
| 97 |
95 96
|
syl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ∩ ran 𝑀 ⊆ ( 𝑀 ‘ 𝑘 ) ) |
| 98 |
97 10
|
sstrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) ∧ 𝑘 ∈ ℕ ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) |
| 99 |
98
|
expcom |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) ) |
| 100 |
93 99
|
vtoclga |
⊢ ( 1 ∈ ℕ → ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) ) |
| 101 |
92 100
|
ax-mp |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ∩ ran 𝑀 ⊆ ∪ 𝐽 ) |
| 102 |
11
|
clsval2 |
⊢ ( ( 𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) ) ) |
| 103 |
91 101 102
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ( ∪ 𝐽 ∖ ∩ ran 𝑀 ) ) ) ) |
| 104 |
90 103
|
eqtr4d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ) = ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) ) |
| 105 |
|
dif0 |
⊢ ( ∪ 𝐽 ∖ ∅ ) = ∪ 𝐽 |
| 106 |
105 86
|
eqtr4id |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∪ 𝐽 ∖ ∅ ) = 𝑋 ) |
| 107 |
104 106
|
eqeq12d |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ( ∪ 𝐽 ∖ ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) ) = ( ∪ 𝐽 ∖ ∅ ) ↔ ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = 𝑋 ) ) |
| 108 |
67 107
|
syl5ib |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = 𝑋 ) ) |
| 109 |
4 108
|
sylan |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ( ( int ‘ 𝐽 ) ‘ ∪ ran ( 𝑥 ∈ ℕ ↦ ( 𝑋 ∖ ( 𝑀 ‘ 𝑥 ) ) ) ) = ∅ → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = 𝑋 ) ) |
| 110 |
45 66 109
|
3syld |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ) → ( ∀ 𝑘 ∈ ℕ ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = 𝑋 ) ) |
| 111 |
110
|
3impia |
⊢ ( ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ∧ 𝑀 : ℕ ⟶ 𝐽 ∧ ∀ 𝑘 ∈ ℕ ( ( cls ‘ 𝐽 ) ‘ ( 𝑀 ‘ 𝑘 ) ) = 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ∩ ran 𝑀 ) = 𝑋 ) |