Metamath Proof Explorer


Theorem cvmlift2lem13

Description: Lemma for cvmlift2 . (Contributed by Mario Carneiro, 7-May-2015)

Ref Expression
Hypotheses cvmlift2.b 𝐵 = 𝐶
cvmlift2.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
cvmlift2.g ( 𝜑𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
cvmlift2.p ( 𝜑𝑃𝐵 )
cvmlift2.i ( 𝜑 → ( 𝐹𝑃 ) = ( 0 𝐺 0 ) )
cvmlift2.h 𝐻 = ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
cvmlift2.k 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑥 ) ) ) ‘ 𝑦 ) )
Assertion cvmlift2lem13 ( 𝜑 → ∃! 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) )

Proof

Step Hyp Ref Expression
1 cvmlift2.b 𝐵 = 𝐶
2 cvmlift2.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
3 cvmlift2.g ( 𝜑𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
4 cvmlift2.p ( 𝜑𝑃𝐵 )
5 cvmlift2.i ( 𝜑 → ( 𝐹𝑃 ) = ( 0 𝐺 0 ) )
6 cvmlift2.h 𝐻 = ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
7 cvmlift2.k 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑥 ) ) ) ‘ 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 = 𝑟𝑑 = 𝑡 ) → 𝑑 = 𝑡 )
16 15 eleq1d ( ( 𝑐 = 𝑟𝑑 = 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 = 𝑟𝑑 = 𝑡 ) → 𝑐 = 𝑟 )
24 23 sneqd ( ( 𝑐 = 𝑟𝑑 = 𝑡 ) → { 𝑐 } = { 𝑟 } )
25 24 fveq2d ( ( 𝑐 = 𝑟𝑑 = 𝑡 ) → ( ( nei ‘ II ) ‘ { 𝑐 } ) = ( ( nei ‘ II ) ‘ { 𝑟 } ) )
26 15 sneqd ( ( 𝑐 = 𝑟𝑑 = 𝑡 ) → { 𝑑 } = { 𝑡 } )
27 26 xpeq2d ( ( 𝑐 = 𝑟𝑑 = 𝑡 ) → ( 𝑢 × { 𝑑 } ) = ( 𝑢 × { 𝑡 } ) )
28 27 sseq1d ( ( 𝑐 = 𝑟𝑑 = 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 = 𝑟𝑑 = 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 = 𝑟𝑑 = 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 = 𝑟𝑑 = 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 = 𝑟𝑑 = 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 ( 𝜑𝐾 ∈ ( ( II ×t II ) Cn 𝐶 ) )
35 1 2 3 4 5 6 7 cvmlift2lem7 ( 𝜑 → ( 𝐹𝐾 ) = 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 ) ∧ ( 𝐹𝐻 ) = ( 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 → ( 𝐹𝑔 ) = ( 𝐹𝐾 ) )
43 42 eqeq1d ( 𝑔 = 𝐾 → ( ( 𝐹𝑔 ) = 𝐺 ↔ ( 𝐹𝐾 ) = 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 → ( ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ( 𝐹𝐾 ) = 𝐺 ∧ ( 0 𝐾 0 ) = 𝑃 ) ) )
47 46 rspcev ( ( 𝐾 ∈ ( ( II ×t II ) Cn 𝐶 ) ∧ ( ( 𝐹𝐾 ) = 𝐺 ∧ ( 0 𝐾 0 ) = 𝑃 ) ) → ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) )
48 34 35 41 47 syl12anc ( 𝜑 → ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 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 → ( 𝐹𝑃 ) = ( 𝐺 ‘ ⟨ 0 , 0 ⟩ ) )
66 1 51 2 55 60 63 3 4 65 cvmliftmo ( 𝜑 → ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 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 ( ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 𝑔 ‘ ⟨ 0 , 0 ⟩ ) = 𝑃 ) )
70 69 rmobii ( ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 𝑔 ‘ ⟨ 0 , 0 ⟩ ) = 𝑃 ) )
71 66 70 sylibr ( 𝜑 → ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) )
72 reu5 ( ∃! 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ↔ ( ∃ 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ∧ ∃* 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) ) )
73 48 71 72 sylanbrc ( 𝜑 → ∃! 𝑔 ∈ ( ( II ×t II ) Cn 𝐶 ) ( ( 𝐹𝑔 ) = 𝐺 ∧ ( 0 𝑔 0 ) = 𝑃 ) )