Step |
Hyp |
Ref |
Expression |
1 |
|
cvmcov.1 |
⊢ 𝑆 = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑢 ∈ 𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢 ∩ 𝑣 ) = ∅ ∧ ( 𝐹 ↾ 𝑢 ) ∈ ( ( 𝐶 ↾t 𝑢 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) |
2 |
|
n0 |
⊢ ( ( 𝑆 ‘ 𝑈 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) |
3 |
|
simpl2 |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝑉 ∈ 𝐽 ) |
4 |
|
simpl1 |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
5 |
|
cvmtop1 |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐶 ∈ Top ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝐶 ∈ Top ) |
7 |
6
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝐶 ∈ Top ) |
8 |
1
|
cvmsss |
⊢ ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → 𝑥 ⊆ 𝐶 ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝑥 ⊆ 𝐶 ) |
10 |
9
|
sselda |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝐶 ) |
11 |
|
cvmcn |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) |
12 |
4 11
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) |
13 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ∧ 𝑉 ∈ 𝐽 ) → ( ◡ 𝐹 “ 𝑉 ) ∈ 𝐶 ) |
14 |
12 3 13
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ◡ 𝐹 “ 𝑉 ) ∈ 𝐶 ) |
15 |
14
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ◡ 𝐹 “ 𝑉 ) ∈ 𝐶 ) |
16 |
|
inopn |
⊢ ( ( 𝐶 ∈ Top ∧ 𝑦 ∈ 𝐶 ∧ ( ◡ 𝐹 “ 𝑉 ) ∈ 𝐶 ) → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ 𝐶 ) |
17 |
7 10 15 16
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ 𝐶 ) |
18 |
17
|
fmpttd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) : 𝑥 ⟶ 𝐶 ) |
19 |
18
|
frnd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝐶 ) |
20 |
1
|
cvmsn0 |
⊢ ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → 𝑥 ≠ ∅ ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝑥 ≠ ∅ ) |
22 |
|
dmmptg |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V → dom ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = 𝑥 ) |
23 |
|
inex1g |
⊢ ( 𝑦 ∈ 𝑥 → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V ) |
24 |
22 23
|
mprg |
⊢ dom ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = 𝑥 |
25 |
24
|
eqeq1i |
⊢ ( dom ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ∅ ↔ 𝑥 = ∅ ) |
26 |
|
dm0rn0 |
⊢ ( dom ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ∅ ↔ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ∅ ) |
27 |
25 26
|
bitr3i |
⊢ ( 𝑥 = ∅ ↔ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ∅ ) |
28 |
27
|
necon3bii |
⊢ ( 𝑥 ≠ ∅ ↔ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ≠ ∅ ) |
29 |
21 28
|
sylib |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ≠ ∅ ) |
30 |
19 29
|
jca |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝐶 ∧ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ≠ ∅ ) ) |
31 |
|
inss2 |
⊢ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑉 ) |
32 |
|
elpw2g |
⊢ ( ( ◡ 𝐹 “ 𝑉 ) ∈ 𝐶 → ( ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ↔ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑉 ) ) ) |
33 |
15 32
|
syl |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ↔ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑉 ) ) ) |
34 |
31 33
|
mpbiri |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ) |
35 |
34
|
fmpttd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) : 𝑥 ⟶ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ) |
36 |
35
|
frnd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ) |
37 |
|
sspwuni |
⊢ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝒫 ( ◡ 𝐹 “ 𝑉 ) ↔ ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ ( ◡ 𝐹 “ 𝑉 ) ) |
38 |
36 37
|
sylib |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ ( ◡ 𝐹 “ 𝑉 ) ) |
39 |
|
simpl3 |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝑉 ⊆ 𝑈 ) |
40 |
|
imass2 |
⊢ ( 𝑉 ⊆ 𝑈 → ( ◡ 𝐹 “ 𝑉 ) ⊆ ( ◡ 𝐹 “ 𝑈 ) ) |
41 |
39 40
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ◡ 𝐹 “ 𝑉 ) ⊆ ( ◡ 𝐹 “ 𝑈 ) ) |
42 |
1
|
cvmsuni |
⊢ ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → ∪ 𝑥 = ( ◡ 𝐹 “ 𝑈 ) ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ∪ 𝑥 = ( ◡ 𝐹 “ 𝑈 ) ) |
44 |
41 43
|
sseqtrrd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ◡ 𝐹 “ 𝑉 ) ⊆ ∪ 𝑥 ) |
45 |
44
|
sselda |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) → 𝑧 ∈ ∪ 𝑥 ) |
46 |
|
eqid |
⊢ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) |
47 |
|
ineq1 |
⊢ ( 𝑦 = 𝑡 → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
48 |
47
|
rspceeqv |
⊢ ( ( 𝑡 ∈ 𝑥 ∧ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
49 |
46 48
|
mpan2 |
⊢ ( 𝑡 ∈ 𝑥 → ∃ 𝑦 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
50 |
49
|
ad2antrl |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
51 |
|
vex |
⊢ 𝑡 ∈ V |
52 |
51
|
inex1 |
⊢ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V |
53 |
|
eqid |
⊢ ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
54 |
53
|
elrnmpt |
⊢ ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ↔ ∃ 𝑦 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
55 |
52 54
|
ax-mp |
⊢ ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ↔ ∃ 𝑦 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
56 |
50 55
|
sylibr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
57 |
|
simprr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → 𝑧 ∈ 𝑡 ) |
58 |
|
simplr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) |
59 |
57 58
|
elind |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → 𝑧 ∈ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
60 |
|
eleq2 |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
61 |
60
|
rspcev |
⊢ ( ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∧ 𝑧 ∈ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) → ∃ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) 𝑧 ∈ 𝑤 ) |
62 |
56 59 61
|
syl2anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ∧ ( 𝑡 ∈ 𝑥 ∧ 𝑧 ∈ 𝑡 ) ) → ∃ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) 𝑧 ∈ 𝑤 ) |
63 |
62
|
rexlimdvaa |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) → ( ∃ 𝑡 ∈ 𝑥 𝑧 ∈ 𝑡 → ∃ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) 𝑧 ∈ 𝑤 ) ) |
64 |
|
eluni2 |
⊢ ( 𝑧 ∈ ∪ 𝑥 ↔ ∃ 𝑡 ∈ 𝑥 𝑧 ∈ 𝑡 ) |
65 |
|
eluni2 |
⊢ ( 𝑧 ∈ ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ↔ ∃ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) 𝑧 ∈ 𝑤 ) |
66 |
63 64 65
|
3imtr4g |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) ) |
67 |
45 66
|
mpd |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑉 ) ) → 𝑧 ∈ ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
68 |
38 67
|
eqelssd |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( ◡ 𝐹 “ 𝑉 ) ) |
69 |
|
eldifsn |
⊢ ( 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ↔ ( 𝑧 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∧ 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
70 |
|
vex |
⊢ 𝑧 ∈ V |
71 |
53
|
elrnmpt |
⊢ ( 𝑧 ∈ V → ( 𝑧 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ↔ ∃ 𝑦 ∈ 𝑥 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
72 |
70 71
|
ax-mp |
⊢ ( 𝑧 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ↔ ∃ 𝑦 ∈ 𝑥 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
73 |
47
|
equcoms |
⊢ ( 𝑡 = 𝑦 → ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
74 |
73
|
necon3ai |
⊢ ( ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ¬ 𝑡 = 𝑦 ) |
75 |
|
simpllr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) |
76 |
|
simplr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑡 ∈ 𝑥 ) |
77 |
|
simpr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑥 ) |
78 |
1
|
cvmsdisj |
⊢ ( ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ∧ 𝑡 ∈ 𝑥 ∧ 𝑦 ∈ 𝑥 ) → ( 𝑡 = 𝑦 ∨ ( 𝑡 ∩ 𝑦 ) = ∅ ) ) |
79 |
75 76 77 78
|
syl3anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑡 = 𝑦 ∨ ( 𝑡 ∩ 𝑦 ) = ∅ ) ) |
80 |
79
|
ord |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → ( ¬ 𝑡 = 𝑦 → ( 𝑡 ∩ 𝑦 ) = ∅ ) ) |
81 |
|
inss1 |
⊢ ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ( 𝑡 ∩ 𝑦 ) |
82 |
|
sseq0 |
⊢ ( ( ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ( 𝑡 ∩ 𝑦 ) ∧ ( 𝑡 ∩ 𝑦 ) = ∅ ) → ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ∅ ) |
83 |
81 82
|
mpan |
⊢ ( ( 𝑡 ∩ 𝑦 ) = ∅ → ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ∅ ) |
84 |
74 80 83
|
syl56 |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ∅ ) ) |
85 |
|
neeq1 |
⊢ ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ↔ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
86 |
|
ineq2 |
⊢ ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
87 |
|
inindir |
⊢ ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
88 |
86 87
|
eqtr4di |
⊢ ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
89 |
88
|
eqeq1d |
⊢ ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ↔ ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ∅ ) ) |
90 |
85 89
|
imbi12d |
⊢ ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ↔ ( ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ 𝑦 ) ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ∅ ) ) ) |
91 |
84 90
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) ) |
92 |
91
|
rexlimdva |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ∃ 𝑦 ∈ 𝑥 𝑧 = ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) ) |
93 |
72 92
|
syl5bi |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝑧 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) ) |
94 |
93
|
impd |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝑧 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∧ 𝑧 ≠ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) |
95 |
69 94
|
syl5bi |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) → ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) |
96 |
95
|
ralrimiv |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) |
97 |
|
inss1 |
⊢ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ 𝑡 |
98 |
|
resabs1 |
⊢ ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ 𝑡 → ( ( 𝐹 ↾ 𝑡 ) ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
99 |
97 98
|
ax-mp |
⊢ ( ( 𝐹 ↾ 𝑡 ) ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
100 |
1
|
cvmshmeo |
⊢ ( ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ 𝑡 ) ∈ ( ( 𝐶 ↾t 𝑡 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) |
101 |
100
|
adantll |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ 𝑡 ) ∈ ( ( 𝐶 ↾t 𝑡 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ) |
102 |
6
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝐶 ∈ Top ) |
103 |
9
|
sselda |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑡 ∈ 𝐶 ) |
104 |
|
elssuni |
⊢ ( 𝑡 ∈ 𝐶 → 𝑡 ⊆ ∪ 𝐶 ) |
105 |
103 104
|
syl |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑡 ⊆ ∪ 𝐶 ) |
106 |
|
eqid |
⊢ ∪ 𝐶 = ∪ 𝐶 |
107 |
106
|
restuni |
⊢ ( ( 𝐶 ∈ Top ∧ 𝑡 ⊆ ∪ 𝐶 ) → 𝑡 = ∪ ( 𝐶 ↾t 𝑡 ) ) |
108 |
102 105 107
|
syl2anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑡 = ∪ ( 𝐶 ↾t 𝑡 ) ) |
109 |
97 108
|
sseqtrid |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ∪ ( 𝐶 ↾t 𝑡 ) ) |
110 |
|
eqid |
⊢ ∪ ( 𝐶 ↾t 𝑡 ) = ∪ ( 𝐶 ↾t 𝑡 ) |
111 |
110
|
hmeores |
⊢ ( ( ( 𝐹 ↾ 𝑡 ) ∈ ( ( 𝐶 ↾t 𝑡 ) Homeo ( 𝐽 ↾t 𝑈 ) ) ∧ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ ∪ ( 𝐶 ↾t 𝑡 ) ) → ( ( 𝐹 ↾ 𝑡 ) ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) ) ) |
112 |
101 109 111
|
syl2anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐹 ↾ 𝑡 ) ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) ) ) |
113 |
99 112
|
eqeltrrid |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) ) ) |
114 |
97
|
a1i |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ 𝑡 ) |
115 |
|
simpr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑡 ∈ 𝑥 ) |
116 |
|
restabs |
⊢ ( ( 𝐶 ∈ Top ∧ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ⊆ 𝑡 ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
117 |
102 114 115 116
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
118 |
|
incom |
⊢ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( ( ◡ 𝐹 “ 𝑉 ) ∩ 𝑡 ) |
119 |
|
cnvresima |
⊢ ( ◡ ( 𝐹 ↾ 𝑡 ) “ 𝑉 ) = ( ( ◡ 𝐹 “ 𝑉 ) ∩ 𝑡 ) |
120 |
118 119
|
eqtr4i |
⊢ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) = ( ◡ ( 𝐹 ↾ 𝑡 ) “ 𝑉 ) |
121 |
120
|
imaeq2i |
⊢ ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( ( 𝐹 ↾ 𝑡 ) “ ( ◡ ( 𝐹 ↾ 𝑡 ) “ 𝑉 ) ) |
122 |
4
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
123 |
|
simplr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) |
124 |
1
|
cvmsf1o |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ 𝑡 ) : 𝑡 –1-1-onto→ 𝑈 ) |
125 |
122 123 115 124
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ 𝑡 ) : 𝑡 –1-1-onto→ 𝑈 ) |
126 |
|
f1ofo |
⊢ ( ( 𝐹 ↾ 𝑡 ) : 𝑡 –1-1-onto→ 𝑈 → ( 𝐹 ↾ 𝑡 ) : 𝑡 –onto→ 𝑈 ) |
127 |
125 126
|
syl |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ 𝑡 ) : 𝑡 –onto→ 𝑈 ) |
128 |
39
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → 𝑉 ⊆ 𝑈 ) |
129 |
|
foimacnv |
⊢ ( ( ( 𝐹 ↾ 𝑡 ) : 𝑡 –onto→ 𝑈 ∧ 𝑉 ⊆ 𝑈 ) → ( ( 𝐹 ↾ 𝑡 ) “ ( ◡ ( 𝐹 ↾ 𝑡 ) “ 𝑉 ) ) = 𝑉 ) |
130 |
127 128 129
|
syl2anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐹 ↾ 𝑡 ) “ ( ◡ ( 𝐹 ↾ 𝑡 ) “ 𝑉 ) ) = 𝑉 ) |
131 |
121 130
|
syl5eq |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = 𝑉 ) |
132 |
131
|
oveq2d |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) = ( ( 𝐽 ↾t 𝑈 ) ↾t 𝑉 ) ) |
133 |
|
cvmtop2 |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐽 ∈ Top ) |
134 |
4 133
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝐽 ∈ Top ) |
135 |
1
|
cvmsrcl |
⊢ ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → 𝑈 ∈ 𝐽 ) |
136 |
135
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → 𝑈 ∈ 𝐽 ) |
137 |
|
restabs |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑉 ⊆ 𝑈 ∧ 𝑈 ∈ 𝐽 ) → ( ( 𝐽 ↾t 𝑈 ) ↾t 𝑉 ) = ( 𝐽 ↾t 𝑉 ) ) |
138 |
134 39 136 137
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ( 𝐽 ↾t 𝑈 ) ↾t 𝑉 ) = ( 𝐽 ↾t 𝑉 ) ) |
139 |
138
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐽 ↾t 𝑈 ) ↾t 𝑉 ) = ( 𝐽 ↾t 𝑉 ) ) |
140 |
132 139
|
eqtrd |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) = ( 𝐽 ↾t 𝑉 ) ) |
141 |
117 140
|
oveq12d |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ( ( 𝐶 ↾t 𝑡 ) ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( ( 𝐽 ↾t 𝑈 ) ↾t ( ( 𝐹 ↾ 𝑡 ) “ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) ) = ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) |
142 |
113 141
|
eleqtrd |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) |
143 |
96 142
|
jca |
⊢ ( ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) ∧ 𝑡 ∈ 𝑥 ) → ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) |
144 |
143
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ∀ 𝑡 ∈ 𝑥 ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) |
145 |
52
|
rgenw |
⊢ ∀ 𝑡 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V |
146 |
47
|
cbvmptv |
⊢ ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( 𝑡 ∈ 𝑥 ↦ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) |
147 |
|
sneq |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → { 𝑤 } = { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) |
148 |
147
|
difeq2d |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) = ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ) |
149 |
|
ineq1 |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝑤 ∩ 𝑧 ) = ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) ) |
150 |
149
|
eqeq1d |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝑤 ∩ 𝑧 ) = ∅ ↔ ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) |
151 |
148 150
|
raleqbidv |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ↔ ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ) ) |
152 |
|
reseq2 |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝐹 ↾ 𝑤 ) = ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
153 |
|
oveq2 |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( 𝐶 ↾t 𝑤 ) = ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ) |
154 |
153
|
oveq1d |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) = ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) |
155 |
152 154
|
eleq12d |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ↔ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) |
156 |
151 155
|
anbi12d |
⊢ ( 𝑤 = ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) → ( ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ↔ ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) ) |
157 |
146 156
|
ralrnmptw |
⊢ ( ∀ 𝑡 ∈ 𝑥 ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∈ V → ( ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ↔ ∀ 𝑡 ∈ 𝑥 ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) ) |
158 |
145 157
|
ax-mp |
⊢ ( ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ↔ ∀ 𝑡 ∈ 𝑥 ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) } ) ( ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( ( 𝐶 ↾t ( 𝑡 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) |
159 |
144 158
|
sylibr |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) |
160 |
68 159
|
jca |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( ◡ 𝐹 “ 𝑉 ) ∧ ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) ) |
161 |
1
|
cvmscbv |
⊢ 𝑆 = ( 𝑎 ∈ 𝐽 ↦ { 𝑏 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑏 = ( ◡ 𝐹 “ 𝑎 ) ∧ ∀ 𝑤 ∈ 𝑏 ( ∀ 𝑧 ∈ ( 𝑏 ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑎 ) ) ) ) } ) |
162 |
161
|
cvmsval |
⊢ ( 𝐶 ∈ Top → ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( 𝑆 ‘ 𝑉 ) ↔ ( 𝑉 ∈ 𝐽 ∧ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝐶 ∧ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ≠ ∅ ) ∧ ( ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( ◡ 𝐹 “ 𝑉 ) ∧ ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) ) ) ) |
163 |
6 162
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( 𝑆 ‘ 𝑉 ) ↔ ( 𝑉 ∈ 𝐽 ∧ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ⊆ 𝐶 ∧ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ≠ ∅ ) ∧ ( ∪ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) = ( ◡ 𝐹 “ 𝑉 ) ∧ ∀ 𝑤 ∈ ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ( ∀ 𝑧 ∈ ( ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∖ { 𝑤 } ) ( 𝑤 ∩ 𝑧 ) = ∅ ∧ ( 𝐹 ↾ 𝑤 ) ∈ ( ( 𝐶 ↾t 𝑤 ) Homeo ( 𝐽 ↾t 𝑉 ) ) ) ) ) ) ) |
164 |
3 30 160 163
|
mpbir3and |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ran ( 𝑦 ∈ 𝑥 ↦ ( 𝑦 ∩ ( ◡ 𝐹 “ 𝑉 ) ) ) ∈ ( 𝑆 ‘ 𝑉 ) ) |
165 |
164
|
ne0d |
⊢ ( ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) ) → ( 𝑆 ‘ 𝑉 ) ≠ ∅ ) |
166 |
165
|
ex |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) → ( 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → ( 𝑆 ‘ 𝑉 ) ≠ ∅ ) ) |
167 |
166
|
exlimdv |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) → ( ∃ 𝑥 𝑥 ∈ ( 𝑆 ‘ 𝑈 ) → ( 𝑆 ‘ 𝑉 ) ≠ ∅ ) ) |
168 |
2 167
|
syl5bi |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈 ) → ( ( 𝑆 ‘ 𝑈 ) ≠ ∅ → ( 𝑆 ‘ 𝑉 ) ≠ ∅ ) ) |