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