Step |
Hyp |
Ref |
Expression |
1 |
|
mretopd.m |
⊢ ( 𝜑 → 𝑀 ∈ ( Moore ‘ 𝐵 ) ) |
2 |
|
mretopd.z |
⊢ ( 𝜑 → ∅ ∈ 𝑀 ) |
3 |
|
mretopd.u |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) → ( 𝑥 ∪ 𝑦 ) ∈ 𝑀 ) |
4 |
|
mretopd.j |
⊢ 𝐽 = { 𝑧 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 } |
5 |
|
unieq |
⊢ ( 𝑎 = ∅ → ∪ 𝑎 = ∪ ∅ ) |
6 |
|
uni0 |
⊢ ∪ ∅ = ∅ |
7 |
5 6
|
eqtrdi |
⊢ ( 𝑎 = ∅ → ∪ 𝑎 = ∅ ) |
8 |
7
|
eleq1d |
⊢ ( 𝑎 = ∅ → ( ∪ 𝑎 ∈ 𝐽 ↔ ∅ ∈ 𝐽 ) ) |
9 |
4
|
ssrab3 |
⊢ 𝐽 ⊆ 𝒫 𝐵 |
10 |
|
sstr |
⊢ ( ( 𝑎 ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝐵 ) → 𝑎 ⊆ 𝒫 𝐵 ) |
11 |
9 10
|
mpan2 |
⊢ ( 𝑎 ⊆ 𝐽 → 𝑎 ⊆ 𝒫 𝐵 ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → 𝑎 ⊆ 𝒫 𝐵 ) |
13 |
|
sspwuni |
⊢ ( 𝑎 ⊆ 𝒫 𝐵 ↔ ∪ 𝑎 ⊆ 𝐵 ) |
14 |
12 13
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ∪ 𝑎 ⊆ 𝐵 ) |
15 |
|
vuniex |
⊢ ∪ 𝑎 ∈ V |
16 |
15
|
elpw |
⊢ ( ∪ 𝑎 ∈ 𝒫 𝐵 ↔ ∪ 𝑎 ⊆ 𝐵 ) |
17 |
14 16
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ∪ 𝑎 ∈ 𝒫 𝐵 ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ∪ 𝑎 ∈ 𝒫 𝐵 ) |
19 |
|
uniiun |
⊢ ∪ 𝑎 = ∪ 𝑏 ∈ 𝑎 𝑏 |
20 |
19
|
difeq2i |
⊢ ( 𝐵 ∖ ∪ 𝑎 ) = ( 𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏 ) |
21 |
|
iindif2 |
⊢ ( 𝑎 ≠ ∅ → ∩ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) = ( 𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏 ) ) |
22 |
21
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ∩ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) = ( 𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏 ) ) |
23 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → 𝑀 ∈ ( Moore ‘ 𝐵 ) ) |
24 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → 𝑎 ≠ ∅ ) |
25 |
|
difeq2 |
⊢ ( 𝑧 = 𝑏 → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ 𝑏 ) ) |
26 |
25
|
eleq1d |
⊢ ( 𝑧 = 𝑏 → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) ) |
27 |
26 4
|
elrab2 |
⊢ ( 𝑏 ∈ 𝐽 ↔ ( 𝑏 ∈ 𝒫 𝐵 ∧ ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) ) |
28 |
27
|
simprbi |
⊢ ( 𝑏 ∈ 𝐽 → ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
29 |
28
|
rgen |
⊢ ∀ 𝑏 ∈ 𝐽 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 |
30 |
|
ssralv |
⊢ ( 𝑎 ⊆ 𝐽 → ( ∀ 𝑏 ∈ 𝐽 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 → ∀ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) ) |
31 |
30
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ( ∀ 𝑏 ∈ 𝐽 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 → ∀ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) ) |
32 |
29 31
|
mpi |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ∀ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
33 |
32
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ∀ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
34 |
|
mreiincl |
⊢ ( ( 𝑀 ∈ ( Moore ‘ 𝐵 ) ∧ 𝑎 ≠ ∅ ∧ ∀ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) → ∩ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
35 |
23 24 33 34
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ∩ 𝑏 ∈ 𝑎 ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
36 |
22 35
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ( 𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏 ) ∈ 𝑀 ) |
37 |
20 36
|
eqeltrid |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ( 𝐵 ∖ ∪ 𝑎 ) ∈ 𝑀 ) |
38 |
|
difeq2 |
⊢ ( 𝑧 = ∪ 𝑎 → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ ∪ 𝑎 ) ) |
39 |
38
|
eleq1d |
⊢ ( 𝑧 = ∪ 𝑎 → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ ∪ 𝑎 ) ∈ 𝑀 ) ) |
40 |
39 4
|
elrab2 |
⊢ ( ∪ 𝑎 ∈ 𝐽 ↔ ( ∪ 𝑎 ∈ 𝒫 𝐵 ∧ ( 𝐵 ∖ ∪ 𝑎 ) ∈ 𝑀 ) ) |
41 |
18 37 40
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) ∧ 𝑎 ≠ ∅ ) → ∪ 𝑎 ∈ 𝐽 ) |
42 |
|
0elpw |
⊢ ∅ ∈ 𝒫 𝐵 |
43 |
42
|
a1i |
⊢ ( 𝜑 → ∅ ∈ 𝒫 𝐵 ) |
44 |
|
mre1cl |
⊢ ( 𝑀 ∈ ( Moore ‘ 𝐵 ) → 𝐵 ∈ 𝑀 ) |
45 |
1 44
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ 𝑀 ) |
46 |
|
difeq2 |
⊢ ( 𝑧 = ∅ → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ ∅ ) ) |
47 |
|
dif0 |
⊢ ( 𝐵 ∖ ∅ ) = 𝐵 |
48 |
46 47
|
eqtrdi |
⊢ ( 𝑧 = ∅ → ( 𝐵 ∖ 𝑧 ) = 𝐵 ) |
49 |
48
|
eleq1d |
⊢ ( 𝑧 = ∅ → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ 𝐵 ∈ 𝑀 ) ) |
50 |
49 4
|
elrab2 |
⊢ ( ∅ ∈ 𝐽 ↔ ( ∅ ∈ 𝒫 𝐵 ∧ 𝐵 ∈ 𝑀 ) ) |
51 |
43 45 50
|
sylanbrc |
⊢ ( 𝜑 → ∅ ∈ 𝐽 ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ∅ ∈ 𝐽 ) |
53 |
8 41 52
|
pm2.61ne |
⊢ ( ( 𝜑 ∧ 𝑎 ⊆ 𝐽 ) → ∪ 𝑎 ∈ 𝐽 ) |
54 |
53
|
ex |
⊢ ( 𝜑 → ( 𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽 ) ) |
55 |
54
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑎 ( 𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽 ) ) |
56 |
|
inss1 |
⊢ ( 𝑎 ∩ 𝑏 ) ⊆ 𝑎 |
57 |
|
difeq2 |
⊢ ( 𝑧 = 𝑎 → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ 𝑎 ) ) |
58 |
57
|
eleq1d |
⊢ ( 𝑧 = 𝑎 → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ) ) |
59 |
58 4
|
elrab2 |
⊢ ( 𝑎 ∈ 𝐽 ↔ ( 𝑎 ∈ 𝒫 𝐵 ∧ ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ) ) |
60 |
59
|
simplbi |
⊢ ( 𝑎 ∈ 𝐽 → 𝑎 ∈ 𝒫 𝐵 ) |
61 |
60
|
elpwid |
⊢ ( 𝑎 ∈ 𝐽 → 𝑎 ⊆ 𝐵 ) |
62 |
61
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → 𝑎 ⊆ 𝐵 ) |
63 |
56 62
|
sstrid |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝑎 ∩ 𝑏 ) ⊆ 𝐵 ) |
64 |
|
vex |
⊢ 𝑎 ∈ V |
65 |
64
|
inex1 |
⊢ ( 𝑎 ∩ 𝑏 ) ∈ V |
66 |
65
|
elpw |
⊢ ( ( 𝑎 ∩ 𝑏 ) ∈ 𝒫 𝐵 ↔ ( 𝑎 ∩ 𝑏 ) ⊆ 𝐵 ) |
67 |
63 66
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝑎 ∩ 𝑏 ) ∈ 𝒫 𝐵 ) |
68 |
|
difindi |
⊢ ( 𝐵 ∖ ( 𝑎 ∩ 𝑏 ) ) = ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) |
69 |
59
|
simprbi |
⊢ ( 𝑎 ∈ 𝐽 → ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ) |
70 |
69
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ) |
71 |
28
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) |
72 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → 𝜑 ) |
73 |
|
uneq1 |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝑎 ) → ( 𝑥 ∪ 𝑦 ) = ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) ) |
74 |
73
|
eleq1d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝑎 ) → ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑀 ↔ ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) ∈ 𝑀 ) ) |
75 |
74
|
imbi2d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝑎 ) → ( ( 𝜑 → ( 𝑥 ∪ 𝑦 ) ∈ 𝑀 ) ↔ ( 𝜑 → ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) ∈ 𝑀 ) ) ) |
76 |
|
uneq2 |
⊢ ( 𝑦 = ( 𝐵 ∖ 𝑏 ) → ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) = ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ) |
77 |
76
|
eleq1d |
⊢ ( 𝑦 = ( 𝐵 ∖ 𝑏 ) → ( ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) ∈ 𝑀 ↔ ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ∈ 𝑀 ) ) |
78 |
77
|
imbi2d |
⊢ ( 𝑦 = ( 𝐵 ∖ 𝑏 ) → ( ( 𝜑 → ( ( 𝐵 ∖ 𝑎 ) ∪ 𝑦 ) ∈ 𝑀 ) ↔ ( 𝜑 → ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ∈ 𝑀 ) ) ) |
79 |
3
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) ) → ( 𝑥 ∪ 𝑦 ) ∈ 𝑀 ) |
80 |
79
|
expcom |
⊢ ( ( 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀 ) → ( 𝜑 → ( 𝑥 ∪ 𝑦 ) ∈ 𝑀 ) ) |
81 |
75 78 80
|
vtocl2ga |
⊢ ( ( ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ∧ ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) → ( 𝜑 → ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ∈ 𝑀 ) ) |
82 |
81
|
imp |
⊢ ( ( ( ( 𝐵 ∖ 𝑎 ) ∈ 𝑀 ∧ ( 𝐵 ∖ 𝑏 ) ∈ 𝑀 ) ∧ 𝜑 ) → ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ∈ 𝑀 ) |
83 |
70 71 72 82
|
syl21anc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( ( 𝐵 ∖ 𝑎 ) ∪ ( 𝐵 ∖ 𝑏 ) ) ∈ 𝑀 ) |
84 |
68 83
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝐵 ∖ ( 𝑎 ∩ 𝑏 ) ) ∈ 𝑀 ) |
85 |
|
difeq2 |
⊢ ( 𝑧 = ( 𝑎 ∩ 𝑏 ) → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ ( 𝑎 ∩ 𝑏 ) ) ) |
86 |
85
|
eleq1d |
⊢ ( 𝑧 = ( 𝑎 ∩ 𝑏 ) → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ ( 𝑎 ∩ 𝑏 ) ) ∈ 𝑀 ) ) |
87 |
86 4
|
elrab2 |
⊢ ( ( 𝑎 ∩ 𝑏 ) ∈ 𝐽 ↔ ( ( 𝑎 ∩ 𝑏 ) ∈ 𝒫 𝐵 ∧ ( 𝐵 ∖ ( 𝑎 ∩ 𝑏 ) ) ∈ 𝑀 ) ) |
88 |
67 84 87
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽 ) ) → ( 𝑎 ∩ 𝑏 ) ∈ 𝐽 ) |
89 |
88
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐽 ∀ 𝑏 ∈ 𝐽 ( 𝑎 ∩ 𝑏 ) ∈ 𝐽 ) |
90 |
45
|
pwexd |
⊢ ( 𝜑 → 𝒫 𝐵 ∈ V ) |
91 |
4 90
|
rabexd |
⊢ ( 𝜑 → 𝐽 ∈ V ) |
92 |
|
istopg |
⊢ ( 𝐽 ∈ V → ( 𝐽 ∈ Top ↔ ( ∀ 𝑎 ( 𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽 ) ∧ ∀ 𝑎 ∈ 𝐽 ∀ 𝑏 ∈ 𝐽 ( 𝑎 ∩ 𝑏 ) ∈ 𝐽 ) ) ) |
93 |
91 92
|
syl |
⊢ ( 𝜑 → ( 𝐽 ∈ Top ↔ ( ∀ 𝑎 ( 𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽 ) ∧ ∀ 𝑎 ∈ 𝐽 ∀ 𝑏 ∈ 𝐽 ( 𝑎 ∩ 𝑏 ) ∈ 𝐽 ) ) ) |
94 |
55 89 93
|
mpbir2and |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
95 |
9
|
unissi |
⊢ ∪ 𝐽 ⊆ ∪ 𝒫 𝐵 |
96 |
|
unipw |
⊢ ∪ 𝒫 𝐵 = 𝐵 |
97 |
95 96
|
sseqtri |
⊢ ∪ 𝐽 ⊆ 𝐵 |
98 |
|
pwidg |
⊢ ( 𝐵 ∈ 𝑀 → 𝐵 ∈ 𝒫 𝐵 ) |
99 |
45 98
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐵 ) |
100 |
|
difid |
⊢ ( 𝐵 ∖ 𝐵 ) = ∅ |
101 |
100 2
|
eqeltrid |
⊢ ( 𝜑 → ( 𝐵 ∖ 𝐵 ) ∈ 𝑀 ) |
102 |
|
difeq2 |
⊢ ( 𝑧 = 𝐵 → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ 𝐵 ) ) |
103 |
102
|
eleq1d |
⊢ ( 𝑧 = 𝐵 → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ 𝐵 ) ∈ 𝑀 ) ) |
104 |
103 4
|
elrab2 |
⊢ ( 𝐵 ∈ 𝐽 ↔ ( 𝐵 ∈ 𝒫 𝐵 ∧ ( 𝐵 ∖ 𝐵 ) ∈ 𝑀 ) ) |
105 |
99 101 104
|
sylanbrc |
⊢ ( 𝜑 → 𝐵 ∈ 𝐽 ) |
106 |
|
unissel |
⊢ ( ( ∪ 𝐽 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽 ) → ∪ 𝐽 = 𝐵 ) |
107 |
97 105 106
|
sylancr |
⊢ ( 𝜑 → ∪ 𝐽 = 𝐵 ) |
108 |
107
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ∪ 𝐽 ) |
109 |
|
istopon |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) ↔ ( 𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽 ) ) |
110 |
94 108 109
|
sylanbrc |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
111 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
112 |
111
|
cldval |
⊢ ( 𝐽 ∈ Top → ( Clsd ‘ 𝐽 ) = { 𝑥 ∈ 𝒫 ∪ 𝐽 ∣ ( ∪ 𝐽 ∖ 𝑥 ) ∈ 𝐽 } ) |
113 |
94 112
|
syl |
⊢ ( 𝜑 → ( Clsd ‘ 𝐽 ) = { 𝑥 ∈ 𝒫 ∪ 𝐽 ∣ ( ∪ 𝐽 ∖ 𝑥 ) ∈ 𝐽 } ) |
114 |
107
|
pweqd |
⊢ ( 𝜑 → 𝒫 ∪ 𝐽 = 𝒫 𝐵 ) |
115 |
107
|
difeq1d |
⊢ ( 𝜑 → ( ∪ 𝐽 ∖ 𝑥 ) = ( 𝐵 ∖ 𝑥 ) ) |
116 |
115
|
eleq1d |
⊢ ( 𝜑 → ( ( ∪ 𝐽 ∖ 𝑥 ) ∈ 𝐽 ↔ ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 ) ) |
117 |
114 116
|
rabeqbidv |
⊢ ( 𝜑 → { 𝑥 ∈ 𝒫 ∪ 𝐽 ∣ ( ∪ 𝐽 ∖ 𝑥 ) ∈ 𝐽 } = { 𝑥 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 } ) |
118 |
4
|
eleq2i |
⊢ ( ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 ↔ ( 𝐵 ∖ 𝑥 ) ∈ { 𝑧 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 } ) |
119 |
|
difss |
⊢ ( 𝐵 ∖ 𝑥 ) ⊆ 𝐵 |
120 |
|
elpw2g |
⊢ ( 𝐵 ∈ 𝑀 → ( ( 𝐵 ∖ 𝑥 ) ∈ 𝒫 𝐵 ↔ ( 𝐵 ∖ 𝑥 ) ⊆ 𝐵 ) ) |
121 |
45 120
|
syl |
⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝑥 ) ∈ 𝒫 𝐵 ↔ ( 𝐵 ∖ 𝑥 ) ⊆ 𝐵 ) ) |
122 |
119 121
|
mpbiri |
⊢ ( 𝜑 → ( 𝐵 ∖ 𝑥 ) ∈ 𝒫 𝐵 ) |
123 |
|
difeq2 |
⊢ ( 𝑧 = ( 𝐵 ∖ 𝑥 ) → ( 𝐵 ∖ 𝑧 ) = ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ) |
124 |
123
|
eleq1d |
⊢ ( 𝑧 = ( 𝐵 ∖ 𝑥 ) → ( ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ) ) |
125 |
124
|
elrab3 |
⊢ ( ( 𝐵 ∖ 𝑥 ) ∈ 𝒫 𝐵 → ( ( 𝐵 ∖ 𝑥 ) ∈ { 𝑧 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 } ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ) ) |
126 |
122 125
|
syl |
⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝑥 ) ∈ { 𝑧 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 } ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ) ) |
127 |
126
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( ( 𝐵 ∖ 𝑥 ) ∈ { 𝑧 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑧 ) ∈ 𝑀 } ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ) ) |
128 |
118 127
|
syl5bb |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ) ) |
129 |
|
elpwi |
⊢ ( 𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵 ) |
130 |
|
dfss4 |
⊢ ( 𝑥 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) = 𝑥 ) |
131 |
129 130
|
sylib |
⊢ ( 𝑥 ∈ 𝒫 𝐵 → ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) = 𝑥 ) |
132 |
131
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) = 𝑥 ) |
133 |
132
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( ( 𝐵 ∖ ( 𝐵 ∖ 𝑥 ) ) ∈ 𝑀 ↔ 𝑥 ∈ 𝑀 ) ) |
134 |
128 133
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 ↔ 𝑥 ∈ 𝑀 ) ) |
135 |
134
|
rabbidva |
⊢ ( 𝜑 → { 𝑥 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 } = { 𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀 } ) |
136 |
|
incom |
⊢ ( 𝑀 ∩ 𝒫 𝐵 ) = ( 𝒫 𝐵 ∩ 𝑀 ) |
137 |
|
dfin5 |
⊢ ( 𝒫 𝐵 ∩ 𝑀 ) = { 𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀 } |
138 |
136 137
|
eqtri |
⊢ ( 𝑀 ∩ 𝒫 𝐵 ) = { 𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀 } |
139 |
|
mresspw |
⊢ ( 𝑀 ∈ ( Moore ‘ 𝐵 ) → 𝑀 ⊆ 𝒫 𝐵 ) |
140 |
1 139
|
syl |
⊢ ( 𝜑 → 𝑀 ⊆ 𝒫 𝐵 ) |
141 |
|
df-ss |
⊢ ( 𝑀 ⊆ 𝒫 𝐵 ↔ ( 𝑀 ∩ 𝒫 𝐵 ) = 𝑀 ) |
142 |
140 141
|
sylib |
⊢ ( 𝜑 → ( 𝑀 ∩ 𝒫 𝐵 ) = 𝑀 ) |
143 |
138 142
|
eqtr3id |
⊢ ( 𝜑 → { 𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀 } = 𝑀 ) |
144 |
135 143
|
eqtrd |
⊢ ( 𝜑 → { 𝑥 ∈ 𝒫 𝐵 ∣ ( 𝐵 ∖ 𝑥 ) ∈ 𝐽 } = 𝑀 ) |
145 |
113 117 144
|
3eqtrrd |
⊢ ( 𝜑 → 𝑀 = ( Clsd ‘ 𝐽 ) ) |
146 |
110 145
|
jca |
⊢ ( 𝜑 → ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) ∧ 𝑀 = ( Clsd ‘ 𝐽 ) ) ) |