Step |
Hyp |
Ref |
Expression |
1 |
|
cvmlift2.b |
⊢ 𝐵 = ∪ 𝐶 |
2 |
|
cvmlift2.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
3 |
|
cvmlift2.g |
⊢ ( 𝜑 → 𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) ) |
4 |
|
cvmlift2.p |
⊢ ( 𝜑 → 𝑃 ∈ 𝐵 ) |
5 |
|
cvmlift2.i |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 0 𝐺 0 ) ) |
6 |
|
cvmlift2.h |
⊢ 𝐻 = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) |
7 |
|
cvmlift2.k |
⊢ 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) |
8 |
|
cvmlift2lem10.s |
⊢ 𝑆 = ( 𝑘 ∈ 𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( ∪ 𝑠 = ( ◡ 𝐹 “ 𝑘 ) ∧ ∀ 𝑐 ∈ 𝑠 ( ∀ 𝑑 ∈ ( 𝑠 ∖ { 𝑐 } ) ( 𝑐 ∩ 𝑑 ) = ∅ ∧ ( 𝐹 ↾ 𝑐 ) ∈ ( ( 𝐶 ↾t 𝑐 ) Homeo ( 𝐽 ↾t 𝑘 ) ) ) ) } ) |
9 |
|
cvmlift2lem9.1 |
⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) ∈ 𝑀 ) |
10 |
|
cvmlift2lem9.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) ) |
11 |
|
cvmlift2lem9.3 |
⊢ ( 𝜑 → 𝑈 ∈ II ) |
12 |
|
cvmlift2lem9.4 |
⊢ ( 𝜑 → 𝑉 ∈ II ) |
13 |
|
cvmlift2lem9.5 |
⊢ ( 𝜑 → ( II ↾t 𝑈 ) ∈ Conn ) |
14 |
|
cvmlift2lem9.6 |
⊢ ( 𝜑 → ( II ↾t 𝑉 ) ∈ Conn ) |
15 |
|
cvmlift2lem9.7 |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
16 |
|
cvmlift2lem9.8 |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
17 |
|
cvmlift2lem9.9 |
⊢ ( 𝜑 → ( 𝑈 × 𝑉 ) ⊆ ( ◡ 𝐺 “ 𝑀 ) ) |
18 |
|
cvmlift2lem9.10 |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
19 |
|
cvmlift2lem9.11 |
⊢ ( 𝜑 → ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn 𝐶 ) ) |
20 |
|
cvmlift2lem9.w |
⊢ 𝑊 = ( ℩ 𝑏 ∈ 𝑇 ( 𝑋 𝐾 𝑌 ) ∈ 𝑏 ) |
21 |
|
iitop |
⊢ II ∈ Top |
22 |
|
iiuni |
⊢ ( 0 [,] 1 ) = ∪ II |
23 |
21 21 22 22
|
txunii |
⊢ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) = ∪ ( II ×t II ) |
24 |
1 2 3 4 5 6 7
|
cvmlift2lem5 |
⊢ ( 𝜑 → 𝐾 : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ 𝐵 ) |
25 |
1 2 3 4 5 6 7
|
cvmlift2lem7 |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐾 ) = 𝐺 ) |
26 |
25 3
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐾 ) ∈ ( ( II ×t II ) Cn 𝐽 ) ) |
27 |
21 21
|
txtopi |
⊢ ( II ×t II ) ∈ Top |
28 |
27
|
a1i |
⊢ ( 𝜑 → ( II ×t II ) ∈ Top ) |
29 |
|
elssuni |
⊢ ( 𝑈 ∈ II → 𝑈 ⊆ ∪ II ) |
30 |
29 22
|
sseqtrrdi |
⊢ ( 𝑈 ∈ II → 𝑈 ⊆ ( 0 [,] 1 ) ) |
31 |
11 30
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ ( 0 [,] 1 ) ) |
32 |
31 15
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( 0 [,] 1 ) ) |
33 |
|
elssuni |
⊢ ( 𝑉 ∈ II → 𝑉 ⊆ ∪ II ) |
34 |
33 22
|
sseqtrrdi |
⊢ ( 𝑉 ∈ II → 𝑉 ⊆ ( 0 [,] 1 ) ) |
35 |
12 34
|
syl |
⊢ ( 𝜑 → 𝑉 ⊆ ( 0 [,] 1 ) ) |
36 |
35 16
|
sseldd |
⊢ ( 𝜑 → 𝑌 ∈ ( 0 [,] 1 ) ) |
37 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ ( 0 [,] 1 ) ∧ 𝑌 ∈ ( 0 [,] 1 ) ) → 〈 𝑋 , 𝑌 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
38 |
32 36 37
|
syl2anc |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
39 |
24 32 36
|
fovrnd |
⊢ ( 𝜑 → ( 𝑋 𝐾 𝑌 ) ∈ 𝐵 ) |
40 |
|
fvco3 |
⊢ ( ( 𝐾 : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ 𝐵 ∧ 〈 𝑋 , 𝑌 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( ( 𝐹 ∘ 𝐾 ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 𝐹 ‘ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ) ) |
41 |
24 38 40
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐾 ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 𝐹 ‘ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ) ) |
42 |
25
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐾 ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 𝐺 ‘ 〈 𝑋 , 𝑌 〉 ) ) |
43 |
41 42
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ) = ( 𝐺 ‘ 〈 𝑋 , 𝑌 〉 ) ) |
44 |
|
df-ov |
⊢ ( 𝑋 𝐾 𝑌 ) = ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) |
45 |
44
|
fveq2i |
⊢ ( 𝐹 ‘ ( 𝑋 𝐾 𝑌 ) ) = ( 𝐹 ‘ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ) |
46 |
|
df-ov |
⊢ ( 𝑋 𝐺 𝑌 ) = ( 𝐺 ‘ 〈 𝑋 , 𝑌 〉 ) |
47 |
43 45 46
|
3eqtr4g |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑋 𝐾 𝑌 ) ) = ( 𝑋 𝐺 𝑌 ) ) |
48 |
47 9
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑋 𝐾 𝑌 ) ) ∈ 𝑀 ) |
49 |
8 1 20
|
cvmsiota |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ ( 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) ∧ ( 𝑋 𝐾 𝑌 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( 𝑋 𝐾 𝑌 ) ) ∈ 𝑀 ) ) → ( 𝑊 ∈ 𝑇 ∧ ( 𝑋 𝐾 𝑌 ) ∈ 𝑊 ) ) |
50 |
2 10 39 48 49
|
syl13anc |
⊢ ( 𝜑 → ( 𝑊 ∈ 𝑇 ∧ ( 𝑋 𝐾 𝑌 ) ∈ 𝑊 ) ) |
51 |
44
|
eleq1i |
⊢ ( ( 𝑋 𝐾 𝑌 ) ∈ 𝑊 ↔ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ∈ 𝑊 ) |
52 |
51
|
anbi2i |
⊢ ( ( 𝑊 ∈ 𝑇 ∧ ( 𝑋 𝐾 𝑌 ) ∈ 𝑊 ) ↔ ( 𝑊 ∈ 𝑇 ∧ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ∈ 𝑊 ) ) |
53 |
50 52
|
sylib |
⊢ ( 𝜑 → ( 𝑊 ∈ 𝑇 ∧ ( 𝐾 ‘ 〈 𝑋 , 𝑌 〉 ) ∈ 𝑊 ) ) |
54 |
|
xpss12 |
⊢ ( ( 𝑈 ⊆ ( 0 [,] 1 ) ∧ 𝑉 ⊆ ( 0 [,] 1 ) ) → ( 𝑈 × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
55 |
31 35 54
|
syl2anc |
⊢ ( 𝜑 → ( 𝑈 × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
56 |
|
snidg |
⊢ ( 𝑚 ∈ 𝑈 → 𝑚 ∈ { 𝑚 } ) |
57 |
56
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑚 ∈ { 𝑚 } ) |
58 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑛 ∈ 𝑉 ) |
59 |
|
ovres |
⊢ ( ( 𝑚 ∈ { 𝑚 } ∧ 𝑛 ∈ 𝑉 ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑛 ) = ( 𝑚 𝐾 𝑛 ) ) |
60 |
57 58 59
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑛 ) = ( 𝑚 𝐾 𝑛 ) ) |
61 |
|
eqid |
⊢ ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) |
62 |
21
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → II ∈ Top ) |
63 |
|
snex |
⊢ { 𝑚 } ∈ V |
64 |
63
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → { 𝑚 } ∈ V ) |
65 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑉 ∈ II ) |
66 |
|
txrest |
⊢ ( ( ( II ∈ Top ∧ II ∈ Top ) ∧ ( { 𝑚 } ∈ V ∧ 𝑉 ∈ II ) ) → ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) = ( ( II ↾t { 𝑚 } ) ×t ( II ↾t 𝑉 ) ) ) |
67 |
62 62 64 65 66
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) = ( ( II ↾t { 𝑚 } ) ×t ( II ↾t 𝑉 ) ) ) |
68 |
|
iitopon |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
69 |
31
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑈 ) → 𝑚 ∈ ( 0 [,] 1 ) ) |
70 |
69
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑚 ∈ ( 0 [,] 1 ) ) |
71 |
|
restsn2 |
⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ 𝑚 ∈ ( 0 [,] 1 ) ) → ( II ↾t { 𝑚 } ) = 𝒫 { 𝑚 } ) |
72 |
68 70 71
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( II ↾t { 𝑚 } ) = 𝒫 { 𝑚 } ) |
73 |
|
pwsn |
⊢ 𝒫 { 𝑚 } = { ∅ , { 𝑚 } } |
74 |
|
indisconn |
⊢ { ∅ , { 𝑚 } } ∈ Conn |
75 |
73 74
|
eqeltri |
⊢ 𝒫 { 𝑚 } ∈ Conn |
76 |
72 75
|
eqeltrdi |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( II ↾t { 𝑚 } ) ∈ Conn ) |
77 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( II ↾t 𝑉 ) ∈ Conn ) |
78 |
|
txconn |
⊢ ( ( ( II ↾t { 𝑚 } ) ∈ Conn ∧ ( II ↾t 𝑉 ) ∈ Conn ) → ( ( II ↾t { 𝑚 } ) ×t ( II ↾t 𝑉 ) ) ∈ Conn ) |
79 |
76 77 78
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( II ↾t { 𝑚 } ) ×t ( II ↾t 𝑉 ) ) ∈ Conn ) |
80 |
67 79
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ∈ Conn ) |
81 |
1 2 3 4 5 6 7
|
cvmlift2lem6 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 [,] 1 ) ) → ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ) |
82 |
70 81
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ) |
83 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑉 ⊆ ( 0 [,] 1 ) ) |
84 |
|
xpss2 |
⊢ ( 𝑉 ⊆ ( 0 [,] 1 ) → ( { 𝑚 } × 𝑉 ) ⊆ ( { 𝑚 } × ( 0 [,] 1 ) ) ) |
85 |
83 84
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × 𝑉 ) ⊆ ( { 𝑚 } × ( 0 [,] 1 ) ) ) |
86 |
70
|
snssd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → { 𝑚 } ⊆ ( 0 [,] 1 ) ) |
87 |
|
xpss1 |
⊢ ( { 𝑚 } ⊆ ( 0 [,] 1 ) → ( { 𝑚 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
88 |
86 87
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
89 |
23
|
restuni |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑚 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( { 𝑚 } × ( 0 [,] 1 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ) |
90 |
27 88 89
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × ( 0 [,] 1 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ) |
91 |
85 90
|
sseqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × 𝑉 ) ⊆ ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ) |
92 |
|
eqid |
⊢ ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) |
93 |
92
|
cnrest |
⊢ ( ( ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ∧ ( { 𝑚 } × 𝑉 ) ⊆ ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ) |
94 |
82 91 93
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ) |
95 |
85
|
resabs1d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑚 } × 𝑉 ) ) = ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ) |
96 |
27
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( II ×t II ) ∈ Top ) |
97 |
|
ovex |
⊢ ( 0 [,] 1 ) ∈ V |
98 |
63 97
|
xpex |
⊢ ( { 𝑚 } × ( 0 [,] 1 ) ) ∈ V |
99 |
98
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × ( 0 [,] 1 ) ) ∈ V ) |
100 |
|
restabs |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑚 } × 𝑉 ) ⊆ ( { 𝑚 } × ( 0 [,] 1 ) ) ∧ ( { 𝑚 } × ( 0 [,] 1 ) ) ∈ V ) → ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑚 } × 𝑉 ) ) = ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ) |
101 |
96 85 99 100
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑚 } × 𝑉 ) ) = ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ) |
102 |
101
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( ( ( II ×t II ) ↾t ( { 𝑚 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) = ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ) |
103 |
94 95 102
|
3eltr3d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ) |
104 |
|
cvmtop1 |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐶 ∈ Top ) |
105 |
2 104
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ Top ) |
106 |
105
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝐶 ∈ Top ) |
107 |
1
|
toptopon |
⊢ ( 𝐶 ∈ Top ↔ 𝐶 ∈ ( TopOn ‘ 𝐵 ) ) |
108 |
106 107
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝐶 ∈ ( TopOn ‘ 𝐵 ) ) |
109 |
|
df-ima |
⊢ ( 𝐾 “ ( { 𝑚 } × 𝑉 ) ) = ran ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) |
110 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑚 ∈ 𝑈 ) |
111 |
110
|
snssd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → { 𝑚 } ⊆ 𝑈 ) |
112 |
|
xpss1 |
⊢ ( { 𝑚 } ⊆ 𝑈 → ( { 𝑚 } × 𝑉 ) ⊆ ( 𝑈 × 𝑉 ) ) |
113 |
|
imass2 |
⊢ ( ( { 𝑚 } × 𝑉 ) ⊆ ( 𝑈 × 𝑉 ) → ( 𝐾 “ ( { 𝑚 } × 𝑉 ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
114 |
111 112 113
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 “ ( { 𝑚 } × 𝑉 ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
115 |
|
imaco |
⊢ ( ( ◡ 𝐾 ∘ ◡ 𝐹 ) “ 𝑀 ) = ( ◡ 𝐾 “ ( ◡ 𝐹 “ 𝑀 ) ) |
116 |
|
cnvco |
⊢ ◡ ( 𝐹 ∘ 𝐾 ) = ( ◡ 𝐾 ∘ ◡ 𝐹 ) |
117 |
25
|
cnveqd |
⊢ ( 𝜑 → ◡ ( 𝐹 ∘ 𝐾 ) = ◡ 𝐺 ) |
118 |
116 117
|
eqtr3id |
⊢ ( 𝜑 → ( ◡ 𝐾 ∘ ◡ 𝐹 ) = ◡ 𝐺 ) |
119 |
118
|
imaeq1d |
⊢ ( 𝜑 → ( ( ◡ 𝐾 ∘ ◡ 𝐹 ) “ 𝑀 ) = ( ◡ 𝐺 “ 𝑀 ) ) |
120 |
115 119
|
eqtr3id |
⊢ ( 𝜑 → ( ◡ 𝐾 “ ( ◡ 𝐹 “ 𝑀 ) ) = ( ◡ 𝐺 “ 𝑀 ) ) |
121 |
17 120
|
sseqtrrd |
⊢ ( 𝜑 → ( 𝑈 × 𝑉 ) ⊆ ( ◡ 𝐾 “ ( ◡ 𝐹 “ 𝑀 ) ) ) |
122 |
24
|
ffund |
⊢ ( 𝜑 → Fun 𝐾 ) |
123 |
24
|
fdmd |
⊢ ( 𝜑 → dom 𝐾 = ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
124 |
55 123
|
sseqtrrd |
⊢ ( 𝜑 → ( 𝑈 × 𝑉 ) ⊆ dom 𝐾 ) |
125 |
|
funimass3 |
⊢ ( ( Fun 𝐾 ∧ ( 𝑈 × 𝑉 ) ⊆ dom 𝐾 ) → ( ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ↔ ( 𝑈 × 𝑉 ) ⊆ ( ◡ 𝐾 “ ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
126 |
122 124 125
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ↔ ( 𝑈 × 𝑉 ) ⊆ ( ◡ 𝐾 “ ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
127 |
121 126
|
mpbird |
⊢ ( 𝜑 → ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
128 |
127
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
129 |
114 128
|
sstrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 “ ( { 𝑚 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
130 |
109 129
|
eqsstrrid |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ran ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
131 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝑀 ) ⊆ dom 𝐹 |
132 |
|
cvmcn |
⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) |
133 |
2 132
|
syl |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ) |
134 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
135 |
1 134
|
cnf |
⊢ ( 𝐹 ∈ ( 𝐶 Cn 𝐽 ) → 𝐹 : 𝐵 ⟶ ∪ 𝐽 ) |
136 |
|
fdm |
⊢ ( 𝐹 : 𝐵 ⟶ ∪ 𝐽 → dom 𝐹 = 𝐵 ) |
137 |
133 135 136
|
3syl |
⊢ ( 𝜑 → dom 𝐹 = 𝐵 ) |
138 |
131 137
|
sseqtrid |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑀 ) ⊆ 𝐵 ) |
139 |
138
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ◡ 𝐹 “ 𝑀 ) ⊆ 𝐵 ) |
140 |
|
cnrest2 |
⊢ ( ( 𝐶 ∈ ( TopOn ‘ 𝐵 ) ∧ ran ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ∧ ( ◡ 𝐹 “ 𝑀 ) ⊆ 𝐵 ) → ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
141 |
108 130 139 140
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
142 |
103 141
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
143 |
8
|
cvmsss |
⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) → 𝑇 ⊆ 𝐶 ) |
144 |
10 143
|
syl |
⊢ ( 𝜑 → 𝑇 ⊆ 𝐶 ) |
145 |
50
|
simpld |
⊢ ( 𝜑 → 𝑊 ∈ 𝑇 ) |
146 |
144 145
|
sseldd |
⊢ ( 𝜑 → 𝑊 ∈ 𝐶 ) |
147 |
|
elssuni |
⊢ ( 𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇 ) |
148 |
145 147
|
syl |
⊢ ( 𝜑 → 𝑊 ⊆ ∪ 𝑇 ) |
149 |
8
|
cvmsuni |
⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) → ∪ 𝑇 = ( ◡ 𝐹 “ 𝑀 ) ) |
150 |
10 149
|
syl |
⊢ ( 𝜑 → ∪ 𝑇 = ( ◡ 𝐹 “ 𝑀 ) ) |
151 |
148 150
|
sseqtrd |
⊢ ( 𝜑 → 𝑊 ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
152 |
8
|
cvmsrcl |
⊢ ( 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) → 𝑀 ∈ 𝐽 ) |
153 |
10 152
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ 𝐽 ) |
154 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( 𝐶 Cn 𝐽 ) ∧ 𝑀 ∈ 𝐽 ) → ( ◡ 𝐹 “ 𝑀 ) ∈ 𝐶 ) |
155 |
133 153 154
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑀 ) ∈ 𝐶 ) |
156 |
|
restopn2 |
⊢ ( ( 𝐶 ∈ Top ∧ ( ◡ 𝐹 “ 𝑀 ) ∈ 𝐶 ) → ( 𝑊 ∈ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ↔ ( 𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
157 |
105 155 156
|
syl2anc |
⊢ ( 𝜑 → ( 𝑊 ∈ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ↔ ( 𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
158 |
146 151 157
|
mpbir2and |
⊢ ( 𝜑 → 𝑊 ∈ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) |
159 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑊 ∈ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) |
160 |
8
|
cvmscld |
⊢ ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ 𝑇 ∈ ( 𝑆 ‘ 𝑀 ) ∧ 𝑊 ∈ 𝑇 ) → 𝑊 ∈ ( Clsd ‘ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
161 |
2 10 145 160
|
syl3anc |
⊢ ( 𝜑 → 𝑊 ∈ ( Clsd ‘ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
162 |
161
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑊 ∈ ( Clsd ‘ ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
163 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑍 ∈ 𝑉 ) |
164 |
|
opelxpi |
⊢ ( ( 𝑚 ∈ { 𝑚 } ∧ 𝑍 ∈ 𝑉 ) → 〈 𝑚 , 𝑍 〉 ∈ ( { 𝑚 } × 𝑉 ) ) |
165 |
57 163 164
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 〈 𝑚 , 𝑍 〉 ∈ ( { 𝑚 } × 𝑉 ) ) |
166 |
85 88
|
sstrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
167 |
23
|
restuni |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑚 } × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( { 𝑚 } × 𝑉 ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ) |
168 |
27 166 167
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( { 𝑚 } × 𝑉 ) = ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ) |
169 |
165 168
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 〈 𝑚 , 𝑍 〉 ∈ ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ) |
170 |
|
df-ov |
⊢ ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑍 ) = ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ‘ 〈 𝑚 , 𝑍 〉 ) |
171 |
|
ovres |
⊢ ( ( 𝑚 ∈ { 𝑚 } ∧ 𝑍 ∈ 𝑉 ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑍 ) = ( 𝑚 𝐾 𝑍 ) ) |
172 |
57 163 171
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑍 ) = ( 𝑚 𝐾 𝑍 ) ) |
173 |
|
snidg |
⊢ ( 𝑍 ∈ 𝑉 → 𝑍 ∈ { 𝑍 } ) |
174 |
18 173
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } ) |
175 |
174
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → 𝑍 ∈ { 𝑍 } ) |
176 |
|
ovres |
⊢ ( ( 𝑚 ∈ 𝑈 ∧ 𝑍 ∈ { 𝑍 } ) → ( 𝑚 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( 𝑚 𝐾 𝑍 ) ) |
177 |
110 175 176
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( 𝑚 𝐾 𝑍 ) ) |
178 |
172 177
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑍 ) = ( 𝑚 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) ) |
179 |
170 178
|
eqtr3id |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ‘ 〈 𝑚 , 𝑍 〉 ) = ( 𝑚 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) ) |
180 |
|
eqid |
⊢ ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) = ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) |
181 |
21
|
a1i |
⊢ ( 𝜑 → II ∈ Top ) |
182 |
|
snex |
⊢ { 𝑍 } ∈ V |
183 |
182
|
a1i |
⊢ ( 𝜑 → { 𝑍 } ∈ V ) |
184 |
|
txrest |
⊢ ( ( ( II ∈ Top ∧ II ∈ Top ) ∧ ( 𝑈 ∈ II ∧ { 𝑍 } ∈ V ) ) → ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) = ( ( II ↾t 𝑈 ) ×t ( II ↾t { 𝑍 } ) ) ) |
185 |
181 181 11 183 184
|
syl22anc |
⊢ ( 𝜑 → ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) = ( ( II ↾t 𝑈 ) ×t ( II ↾t { 𝑍 } ) ) ) |
186 |
35 18
|
sseldd |
⊢ ( 𝜑 → 𝑍 ∈ ( 0 [,] 1 ) ) |
187 |
|
restsn2 |
⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ 𝑍 ∈ ( 0 [,] 1 ) ) → ( II ↾t { 𝑍 } ) = 𝒫 { 𝑍 } ) |
188 |
68 186 187
|
sylancr |
⊢ ( 𝜑 → ( II ↾t { 𝑍 } ) = 𝒫 { 𝑍 } ) |
189 |
|
pwsn |
⊢ 𝒫 { 𝑍 } = { ∅ , { 𝑍 } } |
190 |
|
indisconn |
⊢ { ∅ , { 𝑍 } } ∈ Conn |
191 |
189 190
|
eqeltri |
⊢ 𝒫 { 𝑍 } ∈ Conn |
192 |
188 191
|
eqeltrdi |
⊢ ( 𝜑 → ( II ↾t { 𝑍 } ) ∈ Conn ) |
193 |
|
txconn |
⊢ ( ( ( II ↾t 𝑈 ) ∈ Conn ∧ ( II ↾t { 𝑍 } ) ∈ Conn ) → ( ( II ↾t 𝑈 ) ×t ( II ↾t { 𝑍 } ) ) ∈ Conn ) |
194 |
13 192 193
|
syl2anc |
⊢ ( 𝜑 → ( ( II ↾t 𝑈 ) ×t ( II ↾t { 𝑍 } ) ) ∈ Conn ) |
195 |
185 194
|
eqeltrd |
⊢ ( 𝜑 → ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ∈ Conn ) |
196 |
105 107
|
sylib |
⊢ ( 𝜑 → 𝐶 ∈ ( TopOn ‘ 𝐵 ) ) |
197 |
|
df-ima |
⊢ ( 𝐾 “ ( 𝑈 × { 𝑍 } ) ) = ran ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) |
198 |
18
|
snssd |
⊢ ( 𝜑 → { 𝑍 } ⊆ 𝑉 ) |
199 |
|
xpss2 |
⊢ ( { 𝑍 } ⊆ 𝑉 → ( 𝑈 × { 𝑍 } ) ⊆ ( 𝑈 × 𝑉 ) ) |
200 |
|
imass2 |
⊢ ( ( 𝑈 × { 𝑍 } ) ⊆ ( 𝑈 × 𝑉 ) → ( 𝐾 “ ( 𝑈 × { 𝑍 } ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
201 |
198 199 200
|
3syl |
⊢ ( 𝜑 → ( 𝐾 “ ( 𝑈 × { 𝑍 } ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
202 |
201 127
|
sstrd |
⊢ ( 𝜑 → ( 𝐾 “ ( 𝑈 × { 𝑍 } ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
203 |
197 202
|
eqsstrrid |
⊢ ( 𝜑 → ran ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
204 |
|
cnrest2 |
⊢ ( ( 𝐶 ∈ ( TopOn ‘ 𝐵 ) ∧ ran ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ∧ ( ◡ 𝐹 “ 𝑀 ) ⊆ 𝐵 ) → ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
205 |
196 203 138 204
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
206 |
19 205
|
mpbid |
⊢ ( 𝜑 → ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
207 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑍 ∈ { 𝑍 } ) → 〈 𝑋 , 𝑍 〉 ∈ ( 𝑈 × { 𝑍 } ) ) |
208 |
15 174 207
|
syl2anc |
⊢ ( 𝜑 → 〈 𝑋 , 𝑍 〉 ∈ ( 𝑈 × { 𝑍 } ) ) |
209 |
186
|
snssd |
⊢ ( 𝜑 → { 𝑍 } ⊆ ( 0 [,] 1 ) ) |
210 |
|
xpss12 |
⊢ ( ( 𝑈 ⊆ ( 0 [,] 1 ) ∧ { 𝑍 } ⊆ ( 0 [,] 1 ) ) → ( 𝑈 × { 𝑍 } ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
211 |
31 209 210
|
syl2anc |
⊢ ( 𝜑 → ( 𝑈 × { 𝑍 } ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
212 |
23
|
restuni |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( 𝑈 × { 𝑍 } ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( 𝑈 × { 𝑍 } ) = ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ) |
213 |
27 211 212
|
sylancr |
⊢ ( 𝜑 → ( 𝑈 × { 𝑍 } ) = ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ) |
214 |
208 213
|
eleqtrd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑍 〉 ∈ ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ) |
215 |
|
df-ov |
⊢ ( 𝑋 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ‘ 〈 𝑋 , 𝑍 〉 ) |
216 |
|
ovres |
⊢ ( ( 𝑋 ∈ 𝑈 ∧ 𝑍 ∈ { 𝑍 } ) → ( 𝑋 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( 𝑋 𝐾 𝑍 ) ) |
217 |
15 174 216
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( 𝑋 𝐾 𝑍 ) ) |
218 |
|
snidg |
⊢ ( 𝑋 ∈ 𝑈 → 𝑋 ∈ { 𝑋 } ) |
219 |
15 218
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ { 𝑋 } ) |
220 |
|
ovres |
⊢ ( ( 𝑋 ∈ { 𝑋 } ∧ 𝑍 ∈ 𝑉 ) → ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑍 ) = ( 𝑋 𝐾 𝑍 ) ) |
221 |
219 18 220
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑍 ) = ( 𝑋 𝐾 𝑍 ) ) |
222 |
217 221
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) = ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑍 ) ) |
223 |
215 222
|
eqtr3id |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ‘ 〈 𝑋 , 𝑍 〉 ) = ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑍 ) ) |
224 |
|
eqid |
⊢ ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) |
225 |
|
snex |
⊢ { 𝑋 } ∈ V |
226 |
225
|
a1i |
⊢ ( 𝜑 → { 𝑋 } ∈ V ) |
227 |
|
txrest |
⊢ ( ( ( II ∈ Top ∧ II ∈ Top ) ∧ ( { 𝑋 } ∈ V ∧ 𝑉 ∈ II ) ) → ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) = ( ( II ↾t { 𝑋 } ) ×t ( II ↾t 𝑉 ) ) ) |
228 |
181 181 226 12 227
|
syl22anc |
⊢ ( 𝜑 → ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) = ( ( II ↾t { 𝑋 } ) ×t ( II ↾t 𝑉 ) ) ) |
229 |
|
restsn2 |
⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( II ↾t { 𝑋 } ) = 𝒫 { 𝑋 } ) |
230 |
68 32 229
|
sylancr |
⊢ ( 𝜑 → ( II ↾t { 𝑋 } ) = 𝒫 { 𝑋 } ) |
231 |
|
pwsn |
⊢ 𝒫 { 𝑋 } = { ∅ , { 𝑋 } } |
232 |
|
indisconn |
⊢ { ∅ , { 𝑋 } } ∈ Conn |
233 |
231 232
|
eqeltri |
⊢ 𝒫 { 𝑋 } ∈ Conn |
234 |
230 233
|
eqeltrdi |
⊢ ( 𝜑 → ( II ↾t { 𝑋 } ) ∈ Conn ) |
235 |
|
txconn |
⊢ ( ( ( II ↾t { 𝑋 } ) ∈ Conn ∧ ( II ↾t 𝑉 ) ∈ Conn ) → ( ( II ↾t { 𝑋 } ) ×t ( II ↾t 𝑉 ) ) ∈ Conn ) |
236 |
234 14 235
|
syl2anc |
⊢ ( 𝜑 → ( ( II ↾t { 𝑋 } ) ×t ( II ↾t 𝑉 ) ) ∈ Conn ) |
237 |
228 236
|
eqeltrd |
⊢ ( 𝜑 → ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ∈ Conn ) |
238 |
1 2 3 4 5 6 7
|
cvmlift2lem6 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ) |
239 |
32 238
|
mpdan |
⊢ ( 𝜑 → ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ) |
240 |
|
xpss2 |
⊢ ( 𝑉 ⊆ ( 0 [,] 1 ) → ( { 𝑋 } × 𝑉 ) ⊆ ( { 𝑋 } × ( 0 [,] 1 ) ) ) |
241 |
12 34 240
|
3syl |
⊢ ( 𝜑 → ( { 𝑋 } × 𝑉 ) ⊆ ( { 𝑋 } × ( 0 [,] 1 ) ) ) |
242 |
32
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ ( 0 [,] 1 ) ) |
243 |
|
xpss1 |
⊢ ( { 𝑋 } ⊆ ( 0 [,] 1 ) → ( { 𝑋 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
244 |
242 243
|
syl |
⊢ ( 𝜑 → ( { 𝑋 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
245 |
23
|
restuni |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑋 } × ( 0 [,] 1 ) ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( { 𝑋 } × ( 0 [,] 1 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ) |
246 |
27 244 245
|
sylancr |
⊢ ( 𝜑 → ( { 𝑋 } × ( 0 [,] 1 ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ) |
247 |
241 246
|
sseqtrd |
⊢ ( 𝜑 → ( { 𝑋 } × 𝑉 ) ⊆ ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ) |
248 |
|
eqid |
⊢ ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) |
249 |
248
|
cnrest |
⊢ ( ( ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) Cn 𝐶 ) ∧ ( { 𝑋 } × 𝑉 ) ⊆ ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ) → ( ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ) |
250 |
239 247 249
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ) |
251 |
241
|
resabs1d |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾ ( { 𝑋 } × 𝑉 ) ) = ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ) |
252 |
225 97
|
xpex |
⊢ ( { 𝑋 } × ( 0 [,] 1 ) ) ∈ V |
253 |
252
|
a1i |
⊢ ( 𝜑 → ( { 𝑋 } × ( 0 [,] 1 ) ) ∈ V ) |
254 |
|
restabs |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑋 } × 𝑉 ) ⊆ ( { 𝑋 } × ( 0 [,] 1 ) ) ∧ ( { 𝑋 } × ( 0 [,] 1 ) ) ∈ V ) → ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑋 } × 𝑉 ) ) = ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ) |
255 |
28 241 253 254
|
syl3anc |
⊢ ( 𝜑 → ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑋 } × 𝑉 ) ) = ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ) |
256 |
255
|
oveq1d |
⊢ ( 𝜑 → ( ( ( ( II ×t II ) ↾t ( { 𝑋 } × ( 0 [,] 1 ) ) ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) = ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ) |
257 |
250 251 256
|
3eltr3d |
⊢ ( 𝜑 → ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ) |
258 |
|
df-ima |
⊢ ( 𝐾 “ ( { 𝑋 } × 𝑉 ) ) = ran ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) |
259 |
15
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑈 ) |
260 |
|
xpss1 |
⊢ ( { 𝑋 } ⊆ 𝑈 → ( { 𝑋 } × 𝑉 ) ⊆ ( 𝑈 × 𝑉 ) ) |
261 |
|
imass2 |
⊢ ( ( { 𝑋 } × 𝑉 ) ⊆ ( 𝑈 × 𝑉 ) → ( 𝐾 “ ( { 𝑋 } × 𝑉 ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
262 |
259 260 261
|
3syl |
⊢ ( 𝜑 → ( 𝐾 “ ( { 𝑋 } × 𝑉 ) ) ⊆ ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ) |
263 |
262 127
|
sstrd |
⊢ ( 𝜑 → ( 𝐾 “ ( { 𝑋 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
264 |
258 263
|
eqsstrrid |
⊢ ( 𝜑 → ran ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ) |
265 |
|
cnrest2 |
⊢ ( ( 𝐶 ∈ ( TopOn ‘ 𝐵 ) ∧ ran ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ⊆ ( ◡ 𝐹 “ 𝑀 ) ∧ ( ◡ 𝐹 “ 𝑀 ) ⊆ 𝐵 ) → ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
266 |
196 264 138 265
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn 𝐶 ) ↔ ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) ) |
267 |
257 266
|
mpbid |
⊢ ( 𝜑 → ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) Cn ( 𝐶 ↾t ( ◡ 𝐹 “ 𝑀 ) ) ) ) |
268 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ { 𝑋 } ∧ 𝑌 ∈ 𝑉 ) → 〈 𝑋 , 𝑌 〉 ∈ ( { 𝑋 } × 𝑉 ) ) |
269 |
219 16 268
|
syl2anc |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( { 𝑋 } × 𝑉 ) ) |
270 |
259 260
|
syl |
⊢ ( 𝜑 → ( { 𝑋 } × 𝑉 ) ⊆ ( 𝑈 × 𝑉 ) ) |
271 |
270 55
|
sstrd |
⊢ ( 𝜑 → ( { 𝑋 } × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
272 |
23
|
restuni |
⊢ ( ( ( II ×t II ) ∈ Top ∧ ( { 𝑋 } × 𝑉 ) ⊆ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) → ( { 𝑋 } × 𝑉 ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ) |
273 |
27 271 272
|
sylancr |
⊢ ( 𝜑 → ( { 𝑋 } × 𝑉 ) = ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ) |
274 |
269 273
|
eleqtrd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ) |
275 |
|
df-ov |
⊢ ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑌 ) = ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) |
276 |
|
ovres |
⊢ ( ( 𝑋 ∈ { 𝑋 } ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑌 ) = ( 𝑋 𝐾 𝑌 ) ) |
277 |
219 16 276
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑌 ) = ( 𝑋 𝐾 𝑌 ) ) |
278 |
275 277
|
eqtr3id |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 𝑋 𝐾 𝑌 ) ) |
279 |
50
|
simprd |
⊢ ( 𝜑 → ( 𝑋 𝐾 𝑌 ) ∈ 𝑊 ) |
280 |
278 279
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ∈ 𝑊 ) |
281 |
224 237 267 158 161 274 280
|
conncn |
⊢ ( 𝜑 → ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) : ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ⟶ 𝑊 ) |
282 |
273
|
feq2d |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) : ( { 𝑋 } × 𝑉 ) ⟶ 𝑊 ↔ ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) : ∪ ( ( II ×t II ) ↾t ( { 𝑋 } × 𝑉 ) ) ⟶ 𝑊 ) ) |
283 |
281 282
|
mpbird |
⊢ ( 𝜑 → ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) : ( { 𝑋 } × 𝑉 ) ⟶ 𝑊 ) |
284 |
283 219 18
|
fovrnd |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ↾ ( { 𝑋 } × 𝑉 ) ) 𝑍 ) ∈ 𝑊 ) |
285 |
223 284
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) ‘ 〈 𝑋 , 𝑍 〉 ) ∈ 𝑊 ) |
286 |
180 195 206 158 161 214 285
|
conncn |
⊢ ( 𝜑 → ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) : ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ⟶ 𝑊 ) |
287 |
213
|
feq2d |
⊢ ( 𝜑 → ( ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) : ( 𝑈 × { 𝑍 } ) ⟶ 𝑊 ↔ ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) : ∪ ( ( II ×t II ) ↾t ( 𝑈 × { 𝑍 } ) ) ⟶ 𝑊 ) ) |
288 |
286 287
|
mpbird |
⊢ ( 𝜑 → ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) : ( 𝑈 × { 𝑍 } ) ⟶ 𝑊 ) |
289 |
288
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) : ( 𝑈 × { 𝑍 } ) ⟶ 𝑊 ) |
290 |
289 110 175
|
fovrnd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( 𝑈 × { 𝑍 } ) ) 𝑍 ) ∈ 𝑊 ) |
291 |
179 290
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) ‘ 〈 𝑚 , 𝑍 〉 ) ∈ 𝑊 ) |
292 |
61 80 142 159 162 169 291
|
conncn |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) : ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ⟶ 𝑊 ) |
293 |
168
|
feq2d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) : ( { 𝑚 } × 𝑉 ) ⟶ 𝑊 ↔ ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) : ∪ ( ( II ×t II ) ↾t ( { 𝑚 } × 𝑉 ) ) ⟶ 𝑊 ) ) |
294 |
292 293
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) : ( { 𝑚 } × 𝑉 ) ⟶ 𝑊 ) |
295 |
294 57 58
|
fovrnd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 ( 𝐾 ↾ ( { 𝑚 } × 𝑉 ) ) 𝑛 ) ∈ 𝑊 ) |
296 |
60 295
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ 𝑈 ∧ 𝑛 ∈ 𝑉 ) ) → ( 𝑚 𝐾 𝑛 ) ∈ 𝑊 ) |
297 |
296
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑚 ∈ 𝑈 ∀ 𝑛 ∈ 𝑉 ( 𝑚 𝐾 𝑛 ) ∈ 𝑊 ) |
298 |
|
funimassov |
⊢ ( ( Fun 𝐾 ∧ ( 𝑈 × 𝑉 ) ⊆ dom 𝐾 ) → ( ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ 𝑊 ↔ ∀ 𝑚 ∈ 𝑈 ∀ 𝑛 ∈ 𝑉 ( 𝑚 𝐾 𝑛 ) ∈ 𝑊 ) ) |
299 |
122 124 298
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ 𝑊 ↔ ∀ 𝑚 ∈ 𝑈 ∀ 𝑛 ∈ 𝑉 ( 𝑚 𝐾 𝑛 ) ∈ 𝑊 ) ) |
300 |
297 299
|
mpbird |
⊢ ( 𝜑 → ( 𝐾 “ ( 𝑈 × 𝑉 ) ) ⊆ 𝑊 ) |
301 |
1 23 8 2 24 26 28 38 10 53 55 300
|
cvmlift2lem9a |
⊢ ( 𝜑 → ( 𝐾 ↾ ( 𝑈 × 𝑉 ) ) ∈ ( ( ( II ×t II ) ↾t ( 𝑈 × 𝑉 ) ) Cn 𝐶 ) ) |