Metamath Proof Explorer


Theorem cvmliftlem15

Description: Lemma for cvmlift . Discharge the assumptions of cvmliftlem14 . The set of all open subsets u of the unit interval such that G " u is contained in an even covering of some open set in J is a cover of II by the definition of a covering map, so by the Lebesgue number lemma lebnumii , there is a subdivision of the closed unit interval into N equal parts such that each part is entirely contained within one such open set of J . Then using finite choice ac6sfi to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 . (Contributed by Mario Carneiro, 14-Feb-2015)

Ref Expression
Hypotheses cvmliftlem.1 𝑆 = ( 𝑘𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( 𝑠 = ( 𝐹𝑘 ) ∧ ∀ 𝑢𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢𝑣 ) = ∅ ∧ ( 𝐹𝑢 ) ∈ ( ( 𝐶t 𝑢 ) Homeo ( 𝐽t 𝑘 ) ) ) ) } )
cvmliftlem.b 𝐵 = 𝐶
cvmliftlem.x 𝑋 = 𝐽
cvmliftlem.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
cvmliftlem.g ( 𝜑𝐺 ∈ ( II Cn 𝐽 ) )
cvmliftlem.p ( 𝜑𝑃𝐵 )
cvmliftlem.e ( 𝜑 → ( 𝐹𝑃 ) = ( 𝐺 ‘ 0 ) )
Assertion cvmliftlem15 ( 𝜑 → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )

Proof

Step Hyp Ref Expression
1 cvmliftlem.1 𝑆 = ( 𝑘𝐽 ↦ { 𝑠 ∈ ( 𝒫 𝐶 ∖ { ∅ } ) ∣ ( 𝑠 = ( 𝐹𝑘 ) ∧ ∀ 𝑢𝑠 ( ∀ 𝑣 ∈ ( 𝑠 ∖ { 𝑢 } ) ( 𝑢𝑣 ) = ∅ ∧ ( 𝐹𝑢 ) ∈ ( ( 𝐶t 𝑢 ) Homeo ( 𝐽t 𝑘 ) ) ) ) } )
2 cvmliftlem.b 𝐵 = 𝐶
3 cvmliftlem.x 𝑋 = 𝐽
4 cvmliftlem.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
5 cvmliftlem.g ( 𝜑𝐺 ∈ ( II Cn 𝐽 ) )
6 cvmliftlem.p ( 𝜑𝑃𝐵 )
7 cvmliftlem.e ( 𝜑 → ( 𝐹𝑃 ) = ( 𝐺 ‘ 0 ) )
8 ssrab2 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ II
9 5 ad2antrr ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → 𝐺 ∈ ( II Cn 𝐽 ) )
10 simprl ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → 𝑗𝐽 )
11 cnima ( ( 𝐺 ∈ ( II Cn 𝐽 ) ∧ 𝑗𝐽 ) → ( 𝐺𝑗 ) ∈ II )
12 9 10 11 syl2anc ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝐺𝑗 ) ∈ II )
13 simplr ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → 𝑥 ∈ ( 0 [,] 1 ) )
14 simprrl ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝐺𝑥 ) ∈ 𝑗 )
15 iiuni ( 0 [,] 1 ) = II
16 15 3 cnf ( 𝐺 ∈ ( II Cn 𝐽 ) → 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 )
17 5 16 syl ( 𝜑𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 )
18 17 ad2antrr ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 )
19 ffn ( 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋𝐺 Fn ( 0 [,] 1 ) )
20 elpreima ( 𝐺 Fn ( 0 [,] 1 ) → ( 𝑥 ∈ ( 𝐺𝑗 ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ( 𝐺𝑥 ) ∈ 𝑗 ) ) )
21 18 19 20 3syl ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝑥 ∈ ( 𝐺𝑗 ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ( 𝐺𝑥 ) ∈ 𝑗 ) ) )
22 13 14 21 mpbir2and ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → 𝑥 ∈ ( 𝐺𝑗 ) )
23 simprrr ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝑆𝑗 ) ≠ ∅ )
24 ffun ( 𝐺 : ( 0 [,] 1 ) ⟶ 𝑋 → Fun 𝐺 )
25 funimacnv ( Fun 𝐺 → ( 𝐺 “ ( 𝐺𝑗 ) ) = ( 𝑗 ∩ ran 𝐺 ) )
26 18 24 25 3syl ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝐺 “ ( 𝐺𝑗 ) ) = ( 𝑗 ∩ ran 𝐺 ) )
27 inss1 ( 𝑗 ∩ ran 𝐺 ) ⊆ 𝑗
28 26 27 eqsstrdi ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 )
29 28 ralrimivw ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ∀ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 )
30 r19.2z ( ( ( 𝑆𝑗 ) ≠ ∅ ∧ ∀ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 ) → ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 )
31 23 29 30 syl2anc ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 )
32 eleq2 ( 𝑢 = ( 𝐺𝑗 ) → ( 𝑥𝑢𝑥 ∈ ( 𝐺𝑗 ) ) )
33 imaeq2 ( 𝑢 = ( 𝐺𝑗 ) → ( 𝐺𝑢 ) = ( 𝐺 “ ( 𝐺𝑗 ) ) )
34 33 sseq1d ( 𝑢 = ( 𝐺𝑗 ) → ( ( 𝐺𝑢 ) ⊆ 𝑗 ↔ ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 ) )
35 34 rexbidv ( 𝑢 = ( 𝐺𝑗 ) → ( ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ↔ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 ) )
36 32 35 anbi12d ( 𝑢 = ( 𝐺𝑗 ) → ( ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) ↔ ( 𝑥 ∈ ( 𝐺𝑗 ) ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 ) ) )
37 36 rspcev ( ( ( 𝐺𝑗 ) ∈ II ∧ ( 𝑥 ∈ ( 𝐺𝑗 ) ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺 “ ( 𝐺𝑗 ) ) ⊆ 𝑗 ) ) → ∃ 𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
38 12 22 31 37 syl12anc ( ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) ∧ ( 𝑗𝐽 ∧ ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) ) ) → ∃ 𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
39 4 adantr ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
40 17 ffvelrnda ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) → ( 𝐺𝑥 ) ∈ 𝑋 )
41 1 3 cvmcov ( ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ∧ ( 𝐺𝑥 ) ∈ 𝑋 ) → ∃ 𝑗𝐽 ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) )
42 39 40 41 syl2anc ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) → ∃ 𝑗𝐽 ( ( 𝐺𝑥 ) ∈ 𝑗 ∧ ( 𝑆𝑗 ) ≠ ∅ ) )
43 38 42 reximddv ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) → ∃ 𝑗𝐽𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
44 r19.42v ( ∃ 𝑗𝐽 ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) ↔ ( 𝑥𝑢 ∧ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
45 44 rexbii ( ∃ 𝑢 ∈ II ∃ 𝑗𝐽 ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) ↔ ∃ 𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
46 rexcom ( ∃ 𝑗𝐽𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) ↔ ∃ 𝑢 ∈ II ∃ 𝑗𝐽 ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
47 elunirab ( 𝑥 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ↔ ∃ 𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) )
48 45 46 47 3bitr4i ( ∃ 𝑗𝐽𝑢 ∈ II ( 𝑥𝑢 ∧ ∃ 𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ) ↔ 𝑥 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } )
49 43 48 sylib ( ( 𝜑𝑥 ∈ ( 0 [,] 1 ) ) → 𝑥 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } )
50 49 ex ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) → 𝑥 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ) )
51 50 ssrdv ( 𝜑 → ( 0 [,] 1 ) ⊆ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } )
52 uniss ( { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ II → { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ II )
53 8 52 mp1i ( 𝜑 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ II )
54 53 15 sseqtrrdi ( 𝜑 { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ ( 0 [,] 1 ) )
55 51 54 eqssd ( 𝜑 → ( 0 [,] 1 ) = { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } )
56 lebnumii ( ( { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ⊆ II ∧ ( 0 [,] 1 ) = { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ) → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 )
57 8 55 56 sylancr ( 𝜑 → ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 )
58 fzfi ( 1 ... 𝑛 ) ∈ Fin
59 imaeq2 ( 𝑢 = 𝑣 → ( 𝐺𝑢 ) = ( 𝐺𝑣 ) )
60 59 sseq1d ( 𝑢 = 𝑣 → ( ( 𝐺𝑢 ) ⊆ 𝑗 ↔ ( 𝐺𝑣 ) ⊆ 𝑗 ) )
61 60 2rexbidv ( 𝑢 = 𝑣 → ( ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 ↔ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 ) )
62 61 rexrab ( ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 ↔ ∃ 𝑣 ∈ II ( ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 ∧ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 ) )
63 vex 𝑗 ∈ V
64 vex 𝑠 ∈ V
65 63 64 op1std ( 𝑢 = ⟨ 𝑗 , 𝑠 ⟩ → ( 1st𝑢 ) = 𝑗 )
66 65 sseq2d ( 𝑢 = ⟨ 𝑗 , 𝑠 ⟩ → ( ( 𝐺𝑣 ) ⊆ ( 1st𝑢 ) ↔ ( 𝐺𝑣 ) ⊆ 𝑗 ) )
67 66 rexiunxp ( ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺𝑣 ) ⊆ ( 1st𝑢 ) ↔ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 )
68 imass2 ( ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 𝐺𝑣 ) )
69 sstr2 ( ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 𝐺𝑣 ) → ( ( 𝐺𝑣 ) ⊆ ( 1st𝑢 ) → ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ) )
70 68 69 syl ( ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ( ( 𝐺𝑣 ) ⊆ ( 1st𝑢 ) → ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ) )
71 70 reximdv ( ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ( ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺𝑣 ) ⊆ ( 1st𝑢 ) → ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ) )
72 67 71 syl5bir ( ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ( ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 → ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ) )
73 72 impcom ( ( ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 ∧ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 ) → ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) )
74 73 rexlimivw ( ∃ 𝑣 ∈ II ( ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑣 ) ⊆ 𝑗 ∧ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 ) → ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) )
75 62 74 sylbi ( ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) )
76 75 ralimi ( ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) )
77 fveq2 ( 𝑢 = ( 𝑔𝑘 ) → ( 1st𝑢 ) = ( 1st ‘ ( 𝑔𝑘 ) ) )
78 77 sseq2d ( 𝑢 = ( 𝑔𝑘 ) → ( ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ↔ ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) )
79 78 ac6sfi ( ( ( 1 ... 𝑛 ) ∈ Fin ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑢 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st𝑢 ) ) → ∃ 𝑔 ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) )
80 58 76 79 sylancr ( ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ∃ 𝑔 ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) )
81 4 ad2antrr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
82 5 ad2antrr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝐺 ∈ ( II Cn 𝐽 ) )
83 6 ad2antrr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝑃𝐵 )
84 7 ad2antrr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → ( 𝐹𝑃 ) = ( 𝐺 ‘ 0 ) )
85 simplr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝑛 ∈ ℕ )
86 simprl ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) )
87 sneq ( 𝑗 = 𝑎 → { 𝑗 } = { 𝑎 } )
88 fveq2 ( 𝑗 = 𝑎 → ( 𝑆𝑗 ) = ( 𝑆𝑎 ) )
89 87 88 xpeq12d ( 𝑗 = 𝑎 → ( { 𝑗 } × ( 𝑆𝑗 ) ) = ( { 𝑎 } × ( 𝑆𝑎 ) ) )
90 89 cbviunv 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) = 𝑎𝐽 ( { 𝑎 } × ( 𝑆𝑎 ) )
91 feq3 ( 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) = 𝑎𝐽 ( { 𝑎 } × ( 𝑆𝑎 ) ) → ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ↔ 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑎𝐽 ( { 𝑎 } × ( 𝑆𝑎 ) ) ) )
92 90 91 ax-mp ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ↔ 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑎𝐽 ( { 𝑎 } × ( 𝑆𝑎 ) ) )
93 86 92 sylib ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑎𝐽 ( { 𝑎 } × ( 𝑆𝑎 ) ) )
94 simprr ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) )
95 eqid ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
96 2fveq3 ( 𝑡 = 𝑧 → ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) = ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑧 ) ) )
97 96 cbvmptv ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) = ( 𝑧 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑧 ) ) )
98 eleq2 ( 𝑐 = 𝑏 → ( ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ↔ ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
99 98 cbvriotavw ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) = ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 )
100 fveq1 ( 𝑦 = 𝑥 → ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) = ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) )
101 100 eleq1d ( 𝑦 = 𝑥 → ( ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ↔ ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
102 101 riotabidv ( 𝑦 = 𝑥 → ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) = ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
103 99 102 syl5eq ( 𝑦 = 𝑥 → ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) = ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
104 103 reseq2d ( 𝑦 = 𝑥 → ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) = ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) )
105 104 cnveqd ( 𝑦 = 𝑥 ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) = ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) )
106 105 fveq1d ( 𝑦 = 𝑥 → ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑧 ) ) = ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) )
107 106 mpteq2dv ( 𝑦 = 𝑥 → ( 𝑧 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑧 ) ) ) = ( 𝑧 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) )
108 97 107 syl5eq ( 𝑦 = 𝑥 → ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) = ( 𝑧 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) )
109 oveq1 ( 𝑤 = 𝑚 → ( 𝑤 − 1 ) = ( 𝑚 − 1 ) )
110 109 oveq1d ( 𝑤 = 𝑚 → ( ( 𝑤 − 1 ) / 𝑛 ) = ( ( 𝑚 − 1 ) / 𝑛 ) )
111 oveq1 ( 𝑤 = 𝑚 → ( 𝑤 / 𝑛 ) = ( 𝑚 / 𝑛 ) )
112 110 111 oveq12d ( 𝑤 = 𝑚 → ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) = ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) )
113 2fveq3 ( 𝑤 = 𝑚 → ( 2nd ‘ ( 𝑔𝑤 ) ) = ( 2nd ‘ ( 𝑔𝑚 ) ) )
114 110 fveq2d ( 𝑤 = 𝑚 → ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) = ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) )
115 114 eleq1d ( 𝑤 = 𝑚 → ( ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ↔ ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
116 113 115 riotaeqbidv ( 𝑤 = 𝑚 → ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) = ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) )
117 116 reseq2d ( 𝑤 = 𝑚 → ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) = ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) )
118 117 cnveqd ( 𝑤 = 𝑚 ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) = ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) )
119 118 fveq1d ( 𝑤 = 𝑚 → ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) = ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) )
120 112 119 mpteq12dv ( 𝑤 = 𝑚 → ( 𝑧 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑥 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) = ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) )
121 108 120 cbvmpov ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) = ( 𝑥 ∈ V , 𝑚 ∈ ℕ ↦ ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) )
122 seqeq2 ( ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) = ( 𝑥 ∈ V , 𝑚 ∈ ℕ ↦ ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) ) → seq 0 ( ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) ) = seq 0 ( ( 𝑥 ∈ V , 𝑚 ∈ ℕ ↦ ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) ) )
123 121 122 ax-mp seq 0 ( ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) ) = seq 0 ( ( 𝑥 ∈ V , 𝑚 ∈ ℕ ↦ ( 𝑧 ∈ ( ( ( 𝑚 − 1 ) / 𝑛 ) [,] ( 𝑚 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑏 ∈ ( 2nd ‘ ( 𝑔𝑚 ) ) ( 𝑥 ‘ ( ( 𝑚 − 1 ) / 𝑛 ) ) ∈ 𝑏 ) ) ‘ ( 𝐺𝑧 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) )
124 eqid 𝑘 ∈ ( 1 ... 𝑛 ) ( seq 0 ( ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) ) ‘ 𝑘 ) = 𝑘 ∈ ( 1 ... 𝑛 ) ( seq 0 ( ( 𝑦 ∈ V , 𝑤 ∈ ℕ ↦ ( 𝑡 ∈ ( ( ( 𝑤 − 1 ) / 𝑛 ) [,] ( 𝑤 / 𝑛 ) ) ↦ ( ( 𝐹 ↾ ( 𝑐 ∈ ( 2nd ‘ ( 𝑔𝑤 ) ) ( 𝑦 ‘ ( ( 𝑤 − 1 ) / 𝑛 ) ) ∈ 𝑐 ) ) ‘ ( 𝐺𝑡 ) ) ) ) , ( ( I ↾ ℕ ) ∪ { ⟨ 0 , { ⟨ 0 , 𝑃 ⟩ } ⟩ } ) ) ‘ 𝑘 )
125 1 2 3 81 82 83 84 85 93 94 95 123 124 cvmliftlem14 ( ( ( 𝜑𝑛 ∈ ℕ ) ∧ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) ) → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
126 125 ex ( ( 𝜑𝑛 ∈ ℕ ) → ( ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) )
127 126 exlimdv ( ( 𝜑𝑛 ∈ ℕ ) → ( ∃ 𝑔 ( 𝑔 : ( 1 ... 𝑛 ) ⟶ 𝑗𝐽 ( { 𝑗 } × ( 𝑆𝑗 ) ) ∧ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ( 𝐺 “ ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ) ⊆ ( 1st ‘ ( 𝑔𝑘 ) ) ) → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) )
128 80 127 syl5 ( ( 𝜑𝑛 ∈ ℕ ) → ( ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) )
129 128 rexlimdva ( 𝜑 → ( ∃ 𝑛 ∈ ℕ ∀ 𝑘 ∈ ( 1 ... 𝑛 ) ∃ 𝑣 ∈ { 𝑢 ∈ II ∣ ∃ 𝑗𝐽𝑠 ∈ ( 𝑆𝑗 ) ( 𝐺𝑢 ) ⊆ 𝑗 } ( ( ( 𝑘 − 1 ) / 𝑛 ) [,] ( 𝑘 / 𝑛 ) ) ⊆ 𝑣 → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) )
130 57 129 mpd ( 𝜑 → ∃! 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = 𝐺 ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )