| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cfilfcls.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
|
cfilfcls.2 |
⊢ 𝑋 = dom dom 𝐷 |
| 3 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
| 4 |
3
|
fclselbas |
⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → 𝑥 ∈ ∪ 𝐽 ) |
| 5 |
4
|
adantl |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝑥 ∈ ∪ 𝐽 ) |
| 6 |
|
df-cfil |
⊢ CauFil = ( 𝑑 ∈ ∪ ran ∞Met ↦ { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 7 |
6
|
mptrcl |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → 𝐷 ∈ ∪ ran ∞Met ) |
| 8 |
|
xmetunirn |
⊢ ( 𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ ( ∞Met ‘ dom dom 𝐷 ) ) |
| 9 |
7 8
|
sylib |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → 𝐷 ∈ ( ∞Met ‘ dom dom 𝐷 ) ) |
| 10 |
2
|
fveq2i |
⊢ ( ∞Met ‘ 𝑋 ) = ( ∞Met ‘ dom dom 𝐷 ) |
| 11 |
9 10
|
eleqtrrdi |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 13 |
1
|
mopntopon |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 14 |
12 13
|
syl |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 15 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
| 16 |
14 15
|
syl |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝑋 = ∪ 𝐽 ) |
| 17 |
5 16
|
eleqtrrd |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝑥 ∈ 𝑋 ) |
| 18 |
1
|
mopni2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 19 |
18
|
3expb |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 20 |
12 19
|
sylan |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 21 |
|
cfilfil |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
| 22 |
11 21
|
mpancom |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
| 23 |
22
|
adantr |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
| 24 |
23
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
| 25 |
12
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 26 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) |
| 27 |
|
rphalfcl |
⊢ ( 𝑟 ∈ ℝ+ → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 28 |
27
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 29 |
|
rphalfcl |
⊢ ( ( 𝑟 / 2 ) ∈ ℝ+ → ( ( 𝑟 / 2 ) / 2 ) ∈ ℝ+ ) |
| 30 |
28 29
|
syl |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑟 / 2 ) / 2 ) ∈ ℝ+ ) |
| 31 |
|
cfil3i |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ ( ( 𝑟 / 2 ) / 2 ) ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝑋 ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) |
| 32 |
25 26 30 31
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → ∃ 𝑦 ∈ 𝑋 ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) |
| 33 |
23
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
| 34 |
|
simprr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) |
| 35 |
25
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 36 |
17
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝑥 ∈ 𝑋 ) |
| 37 |
|
rpxr |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ* ) |
| 38 |
37
|
ad2antlr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝑟 ∈ ℝ* ) |
| 39 |
|
blssm |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ) |
| 40 |
35 36 38 39
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ) |
| 41 |
|
simpllr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) |
| 42 |
28
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 43 |
42
|
rpxrd |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑟 / 2 ) ∈ ℝ* ) |
| 44 |
1
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 𝑟 / 2 ) ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∈ 𝐽 ) |
| 45 |
35 36 43 44
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∈ 𝐽 ) |
| 46 |
|
blcntr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 𝑟 / 2 ) ∈ ℝ+ ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 47 |
35 36 42 46
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 48 |
|
fclsopni |
⊢ ( ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ∧ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∈ 𝐽 ∧ 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ≠ ∅ ) |
| 49 |
41 45 47 34 48
|
syl13anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ≠ ∅ ) |
| 50 |
|
n0 |
⊢ ( ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ≠ ∅ ↔ ∃ 𝑧 𝑧 ∈ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) |
| 51 |
49 50
|
sylib |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ∃ 𝑧 𝑧 ∈ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) |
| 52 |
|
elin |
⊢ ( 𝑧 ∈ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ↔ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) |
| 53 |
35
|
adantr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 54 |
|
simplrl |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑦 ∈ 𝑋 ) |
| 55 |
42
|
adantr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 56 |
55
|
rpred |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑟 / 2 ) ∈ ℝ ) |
| 57 |
|
simprr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) |
| 58 |
|
blhalf |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) ∧ ( ( 𝑟 / 2 ) ∈ ℝ ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 59 |
53 54 56 57 58
|
syl22anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 60 |
|
blssm |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 𝑟 / 2 ) ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ 𝑋 ) |
| 61 |
35 36 43 60
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ 𝑋 ) |
| 62 |
61
|
sselda |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → 𝑧 ∈ 𝑋 ) |
| 63 |
62
|
adantrr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑧 ∈ 𝑋 ) |
| 64 |
|
simpllr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑟 ∈ ℝ+ ) |
| 65 |
64
|
rpred |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑟 ∈ ℝ ) |
| 66 |
|
simprl |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 67 |
55
|
rpxrd |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑟 / 2 ) ∈ ℝ* ) |
| 68 |
36
|
adantr |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑥 ∈ 𝑋 ) |
| 69 |
|
blcom |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑟 / 2 ) ∈ ℝ* ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ↔ 𝑥 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 70 |
53 67 68 63 69
|
syl22anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ↔ 𝑥 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 71 |
66 70
|
mpbid |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → 𝑥 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 72 |
|
blhalf |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝑋 ) ∧ ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) → ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 73 |
53 63 65 71 72
|
syl22anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑧 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 74 |
59 73
|
sstrd |
⊢ ( ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) ∧ ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 75 |
74
|
ex |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( ( 𝑧 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∧ 𝑧 ∈ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 76 |
52 75
|
syl5bi |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑧 ∈ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 77 |
76
|
exlimdv |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( ∃ 𝑧 𝑧 ∈ ( ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∩ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 78 |
51 77
|
mpd |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 79 |
|
filss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐹 ) |
| 80 |
33 34 40 78 79
|
syl13anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝑋 ∧ ( 𝑦 ( ball ‘ 𝐷 ) ( ( 𝑟 / 2 ) / 2 ) ) ∈ 𝐹 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐹 ) |
| 81 |
32 80
|
rexlimddv |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐹 ) |
| 82 |
81
|
ad2ant2r |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐹 ) |
| 83 |
|
toponss |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐽 ) → 𝑦 ⊆ 𝑋 ) |
| 84 |
83
|
adantrr |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ⊆ 𝑋 ) |
| 85 |
14 84
|
sylan |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ⊆ 𝑋 ) |
| 86 |
85
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ⊆ 𝑋 ) |
| 87 |
|
simprr |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 88 |
|
filss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋 ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ∈ 𝐹 ) |
| 89 |
24 82 86 87 88
|
syl13anc |
⊢ ( ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ∈ 𝐹 ) |
| 90 |
20 89
|
rexlimddv |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ∈ 𝐹 ) |
| 91 |
90
|
expr |
⊢ ( ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ∧ 𝑦 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) |
| 92 |
91
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) |
| 93 |
|
flimopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
| 94 |
14 23 93
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → ( 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
| 95 |
17 92 94
|
mpbir2and |
⊢ ( ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) → 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ) |
| 96 |
95
|
ex |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ) ) |
| 97 |
96
|
ssrdv |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → ( 𝐽 fClus 𝐹 ) ⊆ ( 𝐽 fLim 𝐹 ) ) |
| 98 |
|
flimfcls |
⊢ ( 𝐽 fLim 𝐹 ) ⊆ ( 𝐽 fClus 𝐹 ) |
| 99 |
98
|
a1i |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → ( 𝐽 fLim 𝐹 ) ⊆ ( 𝐽 fClus 𝐹 ) ) |
| 100 |
97 99
|
eqssd |
⊢ ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) → ( 𝐽 fClus 𝐹 ) = ( 𝐽 fLim 𝐹 ) ) |