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 |
|
fveq2 |
⊢ ( 𝑎 = 𝑧 → ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) = ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑧 ) ) |
9 |
8
|
eleq2d |
⊢ ( 𝑎 = 𝑧 → ( 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) ↔ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑧 ) ) ) |
10 |
9
|
cbvrabv |
⊢ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } = { 𝑧 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑧 ) } |
11 |
|
sneq |
⊢ ( 𝑧 = 𝑏 → { 𝑧 } = { 𝑏 } ) |
12 |
11
|
xpeq2d |
⊢ ( 𝑧 = 𝑏 → ( ( 0 [,] 1 ) × { 𝑧 } ) = ( ( 0 [,] 1 ) × { 𝑏 } ) ) |
13 |
12
|
sseq1d |
⊢ ( 𝑧 = 𝑏 → ( ( ( 0 [,] 1 ) × { 𝑧 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( ( 0 [,] 1 ) × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) |
14 |
13
|
cbvrabv |
⊢ { 𝑧 ∈ ( 0 [,] 1 ) ∣ ( ( 0 [,] 1 ) × { 𝑧 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } } = { 𝑏 ∈ ( 0 [,] 1 ) ∣ ( ( 0 [,] 1 ) × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } } |
15 |
|
simpr |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → 𝑑 = 𝑡 ) |
16 |
15
|
eleq1d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( 𝑑 ∈ ( 0 [,] 1 ) ↔ 𝑡 ∈ ( 0 [,] 1 ) ) ) |
17 |
|
xpeq1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 × { 𝑏 } ) = ( 𝑢 × { 𝑏 } ) ) |
18 |
17
|
sseq1d |
⊢ ( 𝑣 = 𝑢 → ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) |
19 |
|
xpeq1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 × { 𝑑 } ) = ( 𝑢 × { 𝑑 } ) ) |
20 |
19
|
sseq1d |
⊢ ( 𝑣 = 𝑢 → ( ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) |
21 |
18 20
|
bibi12d |
⊢ ( 𝑣 = 𝑢 → ( ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ↔ ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ) |
22 |
21
|
cbvrexvw |
⊢ ( ∃ 𝑣 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ↔ ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) |
23 |
|
simpl |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → 𝑐 = 𝑟 ) |
24 |
23
|
sneqd |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → { 𝑐 } = { 𝑟 } ) |
25 |
24
|
fveq2d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ( nei ‘ II ) ‘ { 𝑐 } ) = ( ( nei ‘ II ) ‘ { 𝑟 } ) ) |
26 |
15
|
sneqd |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → { 𝑑 } = { 𝑡 } ) |
27 |
26
|
xpeq2d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( 𝑢 × { 𝑑 } ) = ( 𝑢 × { 𝑡 } ) ) |
28 |
27
|
sseq1d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) |
29 |
28
|
bibi2d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ↔ ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ) |
30 |
25 29
|
rexeqbidv |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ↔ ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑟 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ) |
31 |
22 30
|
syl5bb |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ∃ 𝑣 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ↔ ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑟 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ) |
32 |
16 31
|
anbi12d |
⊢ ( ( 𝑐 = 𝑟 ∧ 𝑑 = 𝑡 ) → ( ( 𝑑 ∈ ( 0 [,] 1 ) ∧ ∃ 𝑣 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ↔ ( 𝑡 ∈ ( 0 [,] 1 ) ∧ ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑟 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) ) ) |
33 |
32
|
cbvopabv |
⊢ { 〈 𝑐 , 𝑑 〉 ∣ ( 𝑑 ∈ ( 0 [,] 1 ) ∧ ∃ 𝑣 ∈ ( ( nei ‘ II ) ‘ { 𝑐 } ) ( ( 𝑣 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑣 × { 𝑑 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) } = { 〈 𝑟 , 𝑡 〉 ∣ ( 𝑡 ∈ ( 0 [,] 1 ) ∧ ∃ 𝑢 ∈ ( ( nei ‘ II ) ‘ { 𝑟 } ) ( ( 𝑢 × { 𝑏 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ↔ ( 𝑢 × { 𝑡 } ) ⊆ { 𝑎 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ∣ 𝐾 ∈ ( ( ( II ×t II ) CnP 𝐶 ) ‘ 𝑎 ) } ) ) } |
34 |
1 2 3 4 5 6 7 10 14 33
|
cvmlift2lem12 |
⊢ ( 𝜑 → 𝐾 ∈ ( ( II ×t II ) Cn 𝐶 ) ) |
35 |
1 2 3 4 5 6 7
|
cvmlift2lem7 |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐾 ) = 𝐺 ) |
36 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
37 |
1 2 3 4 5 6 7
|
cvmlift2lem8 |
⊢ ( ( 𝜑 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 0 𝐾 0 ) = ( 𝐻 ‘ 0 ) ) |
38 |
36 37
|
mpan2 |
⊢ ( 𝜑 → ( 0 𝐾 0 ) = ( 𝐻 ‘ 0 ) ) |
39 |
1 2 3 4 5 6
|
cvmlift2lem2 |
⊢ ( 𝜑 → ( 𝐻 ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ 𝐻 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝐻 ‘ 0 ) = 𝑃 ) ) |
40 |
39
|
simp3d |
⊢ ( 𝜑 → ( 𝐻 ‘ 0 ) = 𝑃 ) |
41 |
38 40
|
eqtrd |
⊢ ( 𝜑 → ( 0 𝐾 0 ) = 𝑃 ) |
42 |
|
coeq2 |
⊢ ( 𝑔 = 𝐾 → ( 𝐹 ∘ 𝑔 ) = ( 𝐹 ∘ 𝐾 ) ) |
43 |
42
|
eqeq1d |
⊢ ( 𝑔 = 𝐾 → ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ↔ ( 𝐹 ∘ 𝐾 ) = 𝐺 ) ) |
44 |
|
oveq |
⊢ ( 𝑔 = 𝐾 → ( 0 𝑔 0 ) = ( 0 𝐾 0 ) ) |
45 |
44
|
eqeq1d |
⊢ ( 𝑔 = 𝐾 → ( ( 0 𝑔 0 ) = 𝑃 ↔ ( 0 𝐾 0 ) = 𝑃 ) ) |
46 |
43 45
|
anbi12d |
⊢ ( 𝑔 = 𝐾 → ( ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ( 𝐹 ∘ 𝐾 ) = 𝐺 ∧ ( 0 𝐾 0 ) = 𝑃 ) ) ) |
47 |
46
|
rspcev |
⊢ ( ( 𝐾 ∈ ( ( II ×t II ) Cn 𝐶 ) ∧ ( ( 𝐹 ∘ 𝐾 ) = 𝐺 ∧ ( 0 𝐾 0 ) = 𝑃 ) ) → ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) |
48 |
34 35 41 47
|
syl12anc |
⊢ ( 𝜑 → ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) |
49 |
|
iitop |
⊢ II ∈ Top |
50 |
|
iiuni |
⊢ ( 0 [,] 1 ) = ∪ II |
51 |
49 49 50 50
|
txunii |
⊢ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) = ∪ ( II ×t II ) |
52 |
|
iiconn |
⊢ II ∈ Conn |
53 |
|
txconn |
⊢ ( ( II ∈ Conn ∧ II ∈ Conn ) → ( II ×t II ) ∈ Conn ) |
54 |
52 52 53
|
mp2an |
⊢ ( II ×t II ) ∈ Conn |
55 |
54
|
a1i |
⊢ ( 𝜑 → ( II ×t II ) ∈ Conn ) |
56 |
|
iinllyconn |
⊢ II ∈ 𝑛-Locally Conn |
57 |
|
txconn |
⊢ ( ( 𝑥 ∈ Conn ∧ 𝑦 ∈ Conn ) → ( 𝑥 ×t 𝑦 ) ∈ Conn ) |
58 |
57
|
txnlly |
⊢ ( ( II ∈ 𝑛-Locally Conn ∧ II ∈ 𝑛-Locally Conn ) → ( II ×t II ) ∈ 𝑛-Locally Conn ) |
59 |
56 56 58
|
mp2an |
⊢ ( II ×t II ) ∈ 𝑛-Locally Conn |
60 |
59
|
a1i |
⊢ ( 𝜑 → ( II ×t II ) ∈ 𝑛-Locally Conn ) |
61 |
|
opelxpi |
⊢ ( ( 0 ∈ ( 0 [,] 1 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → 〈 0 , 0 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
62 |
36 36 61
|
mp2an |
⊢ 〈 0 , 0 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) |
63 |
62
|
a1i |
⊢ ( 𝜑 → 〈 0 , 0 〉 ∈ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
64 |
|
df-ov |
⊢ ( 0 𝐺 0 ) = ( 𝐺 ‘ 〈 0 , 0 〉 ) |
65 |
5 64
|
eqtrdi |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 〈 0 , 0 〉 ) ) |
66 |
1 51 2 55 60 63 3 4 65
|
cvmliftmo |
⊢ ( 𝜑 → ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 𝑔 ‘ 〈 0 , 0 〉 ) = 𝑃 ) ) |
67 |
|
df-ov |
⊢ ( 0 𝑔 0 ) = ( 𝑔 ‘ 〈 0 , 0 〉 ) |
68 |
67
|
eqeq1i |
⊢ ( ( 0 𝑔 0 ) = 𝑃 ↔ ( 𝑔 ‘ 〈 0 , 0 〉 ) = 𝑃 ) |
69 |
68
|
anbi2i |
⊢ ( ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 𝑔 ‘ 〈 0 , 0 〉 ) = 𝑃 ) ) |
70 |
69
|
rmobii |
⊢ ( ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 𝑔 ‘ 〈 0 , 0 〉 ) = 𝑃 ) ) |
71 |
66 70
|
sylibr |
⊢ ( 𝜑 → ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) |
72 |
|
reu5 |
⊢ ( ∃! 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ∧ ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) ) |
73 |
48 71 72
|
sylanbrc |
⊢ ( 𝜑 → ∃! 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹 ∘ 𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) |