Step |
Hyp |
Ref |
Expression |
1 |
|
qtoprest.2 |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
qtoprest.3 |
⊢ ( 𝜑 → 𝐹 : 𝑋 –onto→ 𝑌 ) |
3 |
|
qtoprest.4 |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑌 ) |
4 |
|
qtoprest.5 |
⊢ ( 𝜑 → 𝐴 = ( ◡ 𝐹 “ 𝑈 ) ) |
5 |
|
qtoprest.6 |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝐽 ∨ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) ) |
6 |
|
fofn |
⊢ ( 𝐹 : 𝑋 –onto→ 𝑌 → 𝐹 Fn 𝑋 ) |
7 |
2 6
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑋 ) |
8 |
|
qtopid |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) |
9 |
1 7 8
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) |
10 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝑈 ) ⊆ dom 𝐹 |
11 |
7
|
fndmd |
⊢ ( 𝜑 → dom 𝐹 = 𝑋 ) |
12 |
10 11
|
sseqtrid |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑈 ) ⊆ 𝑋 ) |
13 |
4 12
|
eqsstrd |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑋 ) |
14 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
15 |
1 14
|
syl |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
16 |
13 15
|
sseqtrd |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝐽 ) |
17 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
18 |
17
|
cnrest |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ∧ 𝐴 ⊆ ∪ 𝐽 ) → ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( 𝐽 qTop 𝐹 ) ) ) |
19 |
9 16 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( 𝐽 qTop 𝐹 ) ) ) |
20 |
|
qtoptopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) ) |
21 |
1 2 20
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) ) |
22 |
|
df-ima |
⊢ ( 𝐹 “ 𝐴 ) = ran ( 𝐹 ↾ 𝐴 ) |
23 |
4
|
imaeq2d |
⊢ ( 𝜑 → ( 𝐹 “ 𝐴 ) = ( 𝐹 “ ( ◡ 𝐹 “ 𝑈 ) ) ) |
24 |
|
foimacnv |
⊢ ( ( 𝐹 : 𝑋 –onto→ 𝑌 ∧ 𝑈 ⊆ 𝑌 ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑈 ) ) = 𝑈 ) |
25 |
2 3 24
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 “ ( ◡ 𝐹 “ 𝑈 ) ) = 𝑈 ) |
26 |
23 25
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 “ 𝐴 ) = 𝑈 ) |
27 |
22 26
|
eqtr3id |
⊢ ( 𝜑 → ran ( 𝐹 ↾ 𝐴 ) = 𝑈 ) |
28 |
|
eqimss |
⊢ ( ran ( 𝐹 ↾ 𝐴 ) = 𝑈 → ran ( 𝐹 ↾ 𝐴 ) ⊆ 𝑈 ) |
29 |
27 28
|
syl |
⊢ ( 𝜑 → ran ( 𝐹 ↾ 𝐴 ) ⊆ 𝑈 ) |
30 |
|
cnrest2 |
⊢ ( ( ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) ∧ ran ( 𝐹 ↾ 𝐴 ) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑌 ) → ( ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( 𝐽 qTop 𝐹 ) ) ↔ ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) ) |
31 |
21 29 3 30
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( 𝐽 qTop 𝐹 ) ) ↔ ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) ) |
32 |
19 31
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
33 |
|
resttopon |
⊢ ( ( ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) ∧ 𝑈 ⊆ 𝑌 ) → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) ) |
34 |
21 3 33
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) ) |
35 |
|
qtopss |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) ∈ ( ( 𝐽 ↾t 𝐴 ) Cn ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ∧ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) ∧ ran ( 𝐹 ↾ 𝐴 ) = 𝑈 ) → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ⊆ ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ) |
36 |
32 34 27 35
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ⊆ ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ) |
37 |
|
resttopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
38 |
1 13 37
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
39 |
|
fnfun |
⊢ ( 𝐹 Fn 𝑋 → Fun 𝐹 ) |
40 |
7 39
|
syl |
⊢ ( 𝜑 → Fun 𝐹 ) |
41 |
13 11
|
sseqtrrd |
⊢ ( 𝜑 → 𝐴 ⊆ dom 𝐹 ) |
42 |
|
fores |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ ( 𝐹 “ 𝐴 ) ) |
43 |
40 41 42
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ ( 𝐹 “ 𝐴 ) ) |
44 |
|
foeq3 |
⊢ ( ( 𝐹 “ 𝐴 ) = 𝑈 → ( ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ ( 𝐹 “ 𝐴 ) ↔ ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ 𝑈 ) ) |
45 |
26 44
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ ( 𝐹 “ 𝐴 ) ↔ ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ 𝑈 ) ) |
46 |
43 45
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ 𝑈 ) |
47 |
|
elqtop3 |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ∧ ( 𝐹 ↾ 𝐴 ) : 𝐴 –onto→ 𝑈 ) → ( 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ↔ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ) |
48 |
38 46 47
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ↔ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ) |
49 |
|
cnvresima |
⊢ ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) = ( ( ◡ 𝐹 “ 𝑥 ) ∩ 𝐴 ) |
50 |
|
imass2 |
⊢ ( 𝑥 ⊆ 𝑈 → ( ◡ 𝐹 “ 𝑥 ) ⊆ ( ◡ 𝐹 “ 𝑈 ) ) |
51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ◡ 𝐹 “ 𝑥 ) ⊆ ( ◡ 𝐹 “ 𝑈 ) ) |
52 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → 𝐴 = ( ◡ 𝐹 “ 𝑈 ) ) |
53 |
51 52
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ◡ 𝐹 “ 𝑥 ) ⊆ 𝐴 ) |
54 |
|
df-ss |
⊢ ( ( ◡ 𝐹 “ 𝑥 ) ⊆ 𝐴 ↔ ( ( ◡ 𝐹 “ 𝑥 ) ∩ 𝐴 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
55 |
53 54
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ( ◡ 𝐹 “ 𝑥 ) ∩ 𝐴 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
56 |
49 55
|
eqtrid |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
57 |
56
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ↔ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) |
58 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑥 ⊆ 𝑈 ) |
59 |
|
df-ss |
⊢ ( 𝑥 ⊆ 𝑈 ↔ ( 𝑥 ∩ 𝑈 ) = 𝑥 ) |
60 |
58 59
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝑥 ∩ 𝑈 ) = 𝑥 ) |
61 |
|
topontop |
⊢ ( ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) → ( 𝐽 qTop 𝐹 ) ∈ Top ) |
62 |
21 61
|
syl |
⊢ ( 𝜑 → ( 𝐽 qTop 𝐹 ) ∈ Top ) |
63 |
62
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝐽 qTop 𝐹 ) ∈ Top ) |
64 |
|
toponmax |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 ∈ 𝐽 ) |
65 |
1 64
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
66 |
|
fornex |
⊢ ( 𝑋 ∈ 𝐽 → ( 𝐹 : 𝑋 –onto→ 𝑌 → 𝑌 ∈ V ) ) |
67 |
65 2 66
|
sylc |
⊢ ( 𝜑 → 𝑌 ∈ V ) |
68 |
67 3
|
ssexd |
⊢ ( 𝜑 → 𝑈 ∈ V ) |
69 |
68
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑈 ∈ V ) |
70 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑈 ⊆ 𝑌 ) |
71 |
58 70
|
sstrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑥 ⊆ 𝑌 ) |
72 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
73 |
1 72
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
74 |
|
restopn2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → ( ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ↔ ( ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑥 ) ⊆ 𝐴 ) ) ) |
75 |
73 74
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝐽 ) → ( ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ↔ ( ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑥 ) ⊆ 𝐴 ) ) ) |
76 |
75
|
simprbda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝐽 ) ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) |
77 |
76
|
adantrl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝐽 ) ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) |
78 |
77
|
an32s |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) |
79 |
|
elqtop3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
80 |
1 2 79
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
81 |
80
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ↔ ( 𝑥 ⊆ 𝑌 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
82 |
71 78 81
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ) |
83 |
|
elrestr |
⊢ ( ( ( 𝐽 qTop 𝐹 ) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ ( 𝐽 qTop 𝐹 ) ) → ( 𝑥 ∩ 𝑈 ) ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
84 |
63 69 82 83
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝑥 ∩ 𝑈 ) ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
85 |
60 84
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ 𝐽 ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
86 |
34
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) ) |
87 |
|
toponuni |
⊢ ( ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) → 𝑈 = ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
88 |
86 87
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑈 = ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
89 |
88
|
difeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑈 ∖ 𝑥 ) = ( ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∖ 𝑥 ) ) |
90 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑈 ⊆ 𝑌 ) |
91 |
21
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) ) |
92 |
|
toponuni |
⊢ ( ( 𝐽 qTop 𝐹 ) ∈ ( TopOn ‘ 𝑌 ) → 𝑌 = ∪ ( 𝐽 qTop 𝐹 ) ) |
93 |
91 92
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑌 = ∪ ( 𝐽 qTop 𝐹 ) ) |
94 |
90 93
|
sseqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑈 ⊆ ∪ ( 𝐽 qTop 𝐹 ) ) |
95 |
90
|
ssdifssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑈 ∖ 𝑥 ) ⊆ 𝑌 ) |
96 |
40
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → Fun 𝐹 ) |
97 |
|
funcnvcnv |
⊢ ( Fun 𝐹 → Fun ◡ ◡ 𝐹 ) |
98 |
|
imadif |
⊢ ( Fun ◡ ◡ 𝐹 → ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) = ( ( ◡ 𝐹 “ 𝑈 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
99 |
96 97 98
|
3syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) = ( ( ◡ 𝐹 “ 𝑈 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
100 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝐴 = ( ◡ 𝐹 “ 𝑈 ) ) |
101 |
100
|
difeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) = ( ( ◡ 𝐹 “ 𝑈 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
102 |
99 101
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) = ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
103 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) |
104 |
38
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
105 |
|
toponuni |
⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
106 |
104 105
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
107 |
106
|
difeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) = ( ∪ ( 𝐽 ↾t 𝐴 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
108 |
|
topontop |
⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) → ( 𝐽 ↾t 𝐴 ) ∈ Top ) |
109 |
104 108
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐽 ↾t 𝐴 ) ∈ Top ) |
110 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) |
111 |
|
eqid |
⊢ ∪ ( 𝐽 ↾t 𝐴 ) = ∪ ( 𝐽 ↾t 𝐴 ) |
112 |
111
|
opncld |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ Top ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) → ( ∪ ( 𝐽 ↾t 𝐴 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ) |
113 |
109 110 112
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ∪ ( 𝐽 ↾t 𝐴 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ) |
114 |
107 113
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ) |
115 |
|
restcldr |
⊢ ( ( 𝐴 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ) → ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
116 |
103 114 115
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
117 |
102 116
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) |
118 |
|
qtopcld |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 : 𝑋 –onto→ 𝑌 ) → ( ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( 𝐽 qTop 𝐹 ) ) ↔ ( ( 𝑈 ∖ 𝑥 ) ⊆ 𝑌 ∧ ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) ) ) |
119 |
1 2 118
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( 𝐽 qTop 𝐹 ) ) ↔ ( ( 𝑈 ∖ 𝑥 ) ⊆ 𝑌 ∧ ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) ) ) |
120 |
119
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( 𝐽 qTop 𝐹 ) ) ↔ ( ( 𝑈 ∖ 𝑥 ) ⊆ 𝑌 ∧ ( ◡ 𝐹 “ ( 𝑈 ∖ 𝑥 ) ) ∈ ( Clsd ‘ 𝐽 ) ) ) ) |
121 |
95 117 120
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( 𝐽 qTop 𝐹 ) ) ) |
122 |
|
difssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑈 ∖ 𝑥 ) ⊆ 𝑈 ) |
123 |
|
eqid |
⊢ ∪ ( 𝐽 qTop 𝐹 ) = ∪ ( 𝐽 qTop 𝐹 ) |
124 |
123
|
restcldi |
⊢ ( ( 𝑈 ⊆ ∪ ( 𝐽 qTop 𝐹 ) ∧ ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( 𝐽 qTop 𝐹 ) ) ∧ ( 𝑈 ∖ 𝑥 ) ⊆ 𝑈 ) → ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
125 |
94 121 122 124
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑈 ∖ 𝑥 ) ∈ ( Clsd ‘ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
126 |
89 125
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∖ 𝑥 ) ∈ ( Clsd ‘ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
127 |
|
topontop |
⊢ ( ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ ( TopOn ‘ 𝑈 ) → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ Top ) |
128 |
86 127
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ Top ) |
129 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑥 ⊆ 𝑈 ) |
130 |
129 88
|
sseqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑥 ⊆ ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
131 |
|
eqid |
⊢ ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) = ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) |
132 |
131
|
isopn2 |
⊢ ( ( ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∈ Top ∧ 𝑥 ⊆ ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) → ( 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ↔ ( ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∖ 𝑥 ) ∈ ( Clsd ‘ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) ) |
133 |
128 130 132
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ↔ ( ∪ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ∖ 𝑥 ) ∈ ( Clsd ‘ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) ) |
134 |
126 133
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) ∧ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
135 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) → ( 𝐴 ∈ 𝐽 ∨ 𝐴 ∈ ( Clsd ‘ 𝐽 ) ) ) |
136 |
85 134 135
|
mpjaodan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑈 ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
137 |
136
|
expr |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
138 |
57 137
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑈 ) → ( ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
139 |
138
|
expimpd |
⊢ ( 𝜑 → ( ( 𝑥 ⊆ 𝑈 ∧ ( ◡ ( 𝐹 ↾ 𝐴 ) “ 𝑥 ) ∈ ( 𝐽 ↾t 𝐴 ) ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
140 |
48 139
|
sylbid |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) → 𝑥 ∈ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) ) |
141 |
140
|
ssrdv |
⊢ ( 𝜑 → ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ⊆ ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) ) |
142 |
36 141
|
eqssd |
⊢ ( 𝜑 → ( ( 𝐽 qTop 𝐹 ) ↾t 𝑈 ) = ( ( 𝐽 ↾t 𝐴 ) qTop ( 𝐹 ↾ 𝐴 ) ) ) |