Step |
Hyp |
Ref |
Expression |
1 |
|
uncmp.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
simpll |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) ) → 𝐽 ∈ Top ) |
3 |
|
simpll |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝐽 ∈ Top ) |
4 |
|
ssun1 |
⊢ 𝑆 ⊆ ( 𝑆 ∪ 𝑇 ) |
5 |
|
sseq2 |
⊢ ( 𝑋 = ( 𝑆 ∪ 𝑇 ) → ( 𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ ( 𝑆 ∪ 𝑇 ) ) ) |
6 |
4 5
|
mpbiri |
⊢ ( 𝑋 = ( 𝑆 ∪ 𝑇 ) → 𝑆 ⊆ 𝑋 ) |
7 |
6
|
ad2antlr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝑆 ⊆ 𝑋 ) |
8 |
1
|
cmpsub |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ↔ ∀ 𝑚 ∈ 𝒫 𝐽 ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) ) |
9 |
3 7 8
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ↔ ∀ 𝑚 ∈ 𝒫 𝐽 ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) ) |
10 |
|
simprr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝑋 = ∪ 𝑐 ) |
11 |
7 10
|
sseqtrd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝑆 ⊆ ∪ 𝑐 ) |
12 |
|
unieq |
⊢ ( 𝑚 = 𝑐 → ∪ 𝑚 = ∪ 𝑐 ) |
13 |
12
|
sseq2d |
⊢ ( 𝑚 = 𝑐 → ( 𝑆 ⊆ ∪ 𝑚 ↔ 𝑆 ⊆ ∪ 𝑐 ) ) |
14 |
|
pweq |
⊢ ( 𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐 ) |
15 |
14
|
ineq1d |
⊢ ( 𝑚 = 𝑐 → ( 𝒫 𝑚 ∩ Fin ) = ( 𝒫 𝑐 ∩ Fin ) ) |
16 |
15
|
rexeqdv |
⊢ ( 𝑚 = 𝑐 → ( ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ↔ ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) |
17 |
13 16
|
imbi12d |
⊢ ( 𝑚 = 𝑐 → ( ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ↔ ( 𝑆 ⊆ ∪ 𝑐 → ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) ) |
18 |
17
|
rspcv |
⊢ ( 𝑐 ∈ 𝒫 𝐽 → ( ∀ 𝑚 ∈ 𝒫 𝐽 ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) → ( 𝑆 ⊆ ∪ 𝑐 → ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) ) |
19 |
18
|
ad2antrl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ∀ 𝑚 ∈ 𝒫 𝐽 ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) → ( 𝑆 ⊆ ∪ 𝑐 → ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) ) |
20 |
11 19
|
mpid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ∀ 𝑚 ∈ 𝒫 𝐽 ( 𝑆 ⊆ ∪ 𝑚 → ∃ 𝑛 ∈ ( 𝒫 𝑚 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) → ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) |
21 |
9 20
|
sylbid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( 𝐽 ↾t 𝑆 ) ∈ Comp → ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ) ) |
22 |
|
ssun2 |
⊢ 𝑇 ⊆ ( 𝑆 ∪ 𝑇 ) |
23 |
|
sseq2 |
⊢ ( 𝑋 = ( 𝑆 ∪ 𝑇 ) → ( 𝑇 ⊆ 𝑋 ↔ 𝑇 ⊆ ( 𝑆 ∪ 𝑇 ) ) ) |
24 |
22 23
|
mpbiri |
⊢ ( 𝑋 = ( 𝑆 ∪ 𝑇 ) → 𝑇 ⊆ 𝑋 ) |
25 |
24
|
ad2antlr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝑇 ⊆ 𝑋 ) |
26 |
1
|
cmpsub |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋 ) → ( ( 𝐽 ↾t 𝑇 ) ∈ Comp ↔ ∀ 𝑟 ∈ 𝒫 𝐽 ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) ) |
27 |
3 25 26
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( 𝐽 ↾t 𝑇 ) ∈ Comp ↔ ∀ 𝑟 ∈ 𝒫 𝐽 ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) ) |
28 |
25 10
|
sseqtrd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → 𝑇 ⊆ ∪ 𝑐 ) |
29 |
|
unieq |
⊢ ( 𝑟 = 𝑐 → ∪ 𝑟 = ∪ 𝑐 ) |
30 |
29
|
sseq2d |
⊢ ( 𝑟 = 𝑐 → ( 𝑇 ⊆ ∪ 𝑟 ↔ 𝑇 ⊆ ∪ 𝑐 ) ) |
31 |
|
pweq |
⊢ ( 𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐 ) |
32 |
31
|
ineq1d |
⊢ ( 𝑟 = 𝑐 → ( 𝒫 𝑟 ∩ Fin ) = ( 𝒫 𝑐 ∩ Fin ) ) |
33 |
32
|
rexeqdv |
⊢ ( 𝑟 = 𝑐 → ( ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ↔ ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) |
34 |
30 33
|
imbi12d |
⊢ ( 𝑟 = 𝑐 → ( ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ↔ ( 𝑇 ⊆ ∪ 𝑐 → ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) ) |
35 |
34
|
rspcv |
⊢ ( 𝑐 ∈ 𝒫 𝐽 → ( ∀ 𝑟 ∈ 𝒫 𝐽 ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) → ( 𝑇 ⊆ ∪ 𝑐 → ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) ) |
36 |
35
|
ad2antrl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ∀ 𝑟 ∈ 𝒫 𝐽 ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) → ( 𝑇 ⊆ ∪ 𝑐 → ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) ) |
37 |
28 36
|
mpid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ∀ 𝑟 ∈ 𝒫 𝐽 ( 𝑇 ⊆ ∪ 𝑟 → ∃ 𝑠 ∈ ( 𝒫 𝑟 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) → ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) |
38 |
27 37
|
sylbid |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( 𝐽 ↾t 𝑇 ) ∈ Comp → ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) |
39 |
|
reeanv |
⊢ ( ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ↔ ( ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ∧ ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) ) |
40 |
|
elinel1 |
⊢ ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑛 ∈ 𝒫 𝑐 ) |
41 |
40
|
elpwid |
⊢ ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑛 ⊆ 𝑐 ) |
42 |
|
elinel1 |
⊢ ( 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑠 ∈ 𝒫 𝑐 ) |
43 |
42
|
elpwid |
⊢ ( 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑠 ⊆ 𝑐 ) |
44 |
41 43
|
anim12i |
⊢ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) → ( 𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐 ) ) |
45 |
44
|
ad2antrl |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐 ) ) |
46 |
|
unss |
⊢ ( ( 𝑛 ⊆ 𝑐 ∧ 𝑠 ⊆ 𝑐 ) ↔ ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ) |
47 |
45 46
|
sylib |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ) |
48 |
|
elinel2 |
⊢ ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑛 ∈ Fin ) |
49 |
|
elinel2 |
⊢ ( 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) → 𝑠 ∈ Fin ) |
50 |
|
unfi |
⊢ ( ( 𝑛 ∈ Fin ∧ 𝑠 ∈ Fin ) → ( 𝑛 ∪ 𝑠 ) ∈ Fin ) |
51 |
48 49 50
|
syl2an |
⊢ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) → ( 𝑛 ∪ 𝑠 ) ∈ Fin ) |
52 |
51
|
ad2antrl |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑛 ∪ 𝑠 ) ∈ Fin ) |
53 |
47 52
|
jca |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ∧ ( 𝑛 ∪ 𝑠 ) ∈ Fin ) ) |
54 |
|
elin |
⊢ ( ( 𝑛 ∪ 𝑠 ) ∈ ( 𝒫 𝑐 ∩ Fin ) ↔ ( ( 𝑛 ∪ 𝑠 ) ∈ 𝒫 𝑐 ∧ ( 𝑛 ∪ 𝑠 ) ∈ Fin ) ) |
55 |
|
vex |
⊢ 𝑐 ∈ V |
56 |
55
|
elpw2 |
⊢ ( ( 𝑛 ∪ 𝑠 ) ∈ 𝒫 𝑐 ↔ ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ) |
57 |
56
|
anbi1i |
⊢ ( ( ( 𝑛 ∪ 𝑠 ) ∈ 𝒫 𝑐 ∧ ( 𝑛 ∪ 𝑠 ) ∈ Fin ) ↔ ( ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ∧ ( 𝑛 ∪ 𝑠 ) ∈ Fin ) ) |
58 |
54 57
|
bitr2i |
⊢ ( ( ( 𝑛 ∪ 𝑠 ) ⊆ 𝑐 ∧ ( 𝑛 ∪ 𝑠 ) ∈ Fin ) ↔ ( 𝑛 ∪ 𝑠 ) ∈ ( 𝒫 𝑐 ∩ Fin ) ) |
59 |
53 58
|
sylib |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑛 ∪ 𝑠 ) ∈ ( 𝒫 𝑐 ∩ Fin ) ) |
60 |
|
simpllr |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → 𝑋 = ( 𝑆 ∪ 𝑇 ) ) |
61 |
|
ssun3 |
⊢ ( 𝑆 ⊆ ∪ 𝑛 → 𝑆 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) |
62 |
|
ssun4 |
⊢ ( 𝑇 ⊆ ∪ 𝑠 → 𝑇 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) |
63 |
61 62
|
anim12i |
⊢ ( ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) → ( 𝑆 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ∧ 𝑇 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) ) |
64 |
63
|
ad2antll |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑆 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ∧ 𝑇 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) ) |
65 |
|
unss |
⊢ ( ( 𝑆 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ∧ 𝑇 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) ↔ ( 𝑆 ∪ 𝑇 ) ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) |
66 |
64 65
|
sylib |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑆 ∪ 𝑇 ) ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) |
67 |
60 66
|
eqsstrd |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → 𝑋 ⊆ ( ∪ 𝑛 ∪ ∪ 𝑠 ) ) |
68 |
|
uniun |
⊢ ∪ ( 𝑛 ∪ 𝑠 ) = ( ∪ 𝑛 ∪ ∪ 𝑠 ) |
69 |
67 68
|
sseqtrrdi |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → 𝑋 ⊆ ∪ ( 𝑛 ∪ 𝑠 ) ) |
70 |
|
elpwi |
⊢ ( 𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽 ) |
71 |
70
|
adantr |
⊢ ( ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) → 𝑐 ⊆ 𝐽 ) |
72 |
71
|
ad2antlr |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → 𝑐 ⊆ 𝐽 ) |
73 |
47 72
|
sstrd |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ( 𝑛 ∪ 𝑠 ) ⊆ 𝐽 ) |
74 |
|
uniss |
⊢ ( ( 𝑛 ∪ 𝑠 ) ⊆ 𝐽 → ∪ ( 𝑛 ∪ 𝑠 ) ⊆ ∪ 𝐽 ) |
75 |
74 1
|
sseqtrrdi |
⊢ ( ( 𝑛 ∪ 𝑠 ) ⊆ 𝐽 → ∪ ( 𝑛 ∪ 𝑠 ) ⊆ 𝑋 ) |
76 |
73 75
|
syl |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ∪ ( 𝑛 ∪ 𝑠 ) ⊆ 𝑋 ) |
77 |
69 76
|
eqssd |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → 𝑋 = ∪ ( 𝑛 ∪ 𝑠 ) ) |
78 |
|
unieq |
⊢ ( 𝑑 = ( 𝑛 ∪ 𝑠 ) → ∪ 𝑑 = ∪ ( 𝑛 ∪ 𝑠 ) ) |
79 |
78
|
rspceeqv |
⊢ ( ( ( 𝑛 ∪ 𝑠 ) ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑋 = ∪ ( 𝑛 ∪ 𝑠 ) ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) |
80 |
59 77 79
|
syl2anc |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) ∧ ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) ∧ ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) ) ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) |
81 |
80
|
exp32 |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∧ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ) → ( ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) ) |
82 |
81
|
rexlimdvv |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) ( 𝑆 ⊆ ∪ 𝑛 ∧ 𝑇 ⊆ ∪ 𝑠 ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) |
83 |
39 82
|
syl5bir |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( ∃ 𝑛 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑆 ⊆ ∪ 𝑛 ∧ ∃ 𝑠 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑇 ⊆ ∪ 𝑠 ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) |
84 |
21 38 83
|
syl2and |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) ) → ( ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) |
85 |
84
|
impancom |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) ) → ( ( 𝑐 ∈ 𝒫 𝐽 ∧ 𝑋 = ∪ 𝑐 ) → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) |
86 |
85
|
expd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) ) → ( 𝑐 ∈ 𝒫 𝐽 → ( 𝑋 = ∪ 𝑐 → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) ) |
87 |
86
|
ralrimiv |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) ) → ∀ 𝑐 ∈ 𝒫 𝐽 ( 𝑋 = ∪ 𝑐 → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) |
88 |
1
|
iscmp |
⊢ ( 𝐽 ∈ Comp ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑐 ∈ 𝒫 𝐽 ( 𝑋 = ∪ 𝑐 → ∃ 𝑑 ∈ ( 𝒫 𝑐 ∩ Fin ) 𝑋 = ∪ 𝑑 ) ) ) |
89 |
2 87 88
|
sylanbrc |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝑋 = ( 𝑆 ∪ 𝑇 ) ) ∧ ( ( 𝐽 ↾t 𝑆 ) ∈ Comp ∧ ( 𝐽 ↾t 𝑇 ) ∈ Comp ) ) → 𝐽 ∈ Comp ) |