Step | Hyp | Ref
| Expression |
1 | | fzofi 13426 |
. . . . . 6
⊢
(0..^𝐾) ∈
Fin |
2 | | fzfi 13424 |
. . . . . 6
⊢
(1...𝑁) ∈
Fin |
3 | | mapfi 8886 |
. . . . . 6
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0..^𝐾)
↑m (1...𝑁))
∈ Fin) |
4 | 1, 2, 3 | mp2an 692 |
. . . . 5
⊢
((0..^𝐾)
↑m (1...𝑁))
∈ Fin |
5 | | mapfi 8886 |
. . . . . . 7
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑m (1...𝑁))
∈ Fin) |
6 | 2, 2, 5 | mp2an 692 |
. . . . . 6
⊢
((1...𝑁)
↑m (1...𝑁))
∈ Fin |
7 | | f1of 6612 |
. . . . . . . 8
⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) |
8 | 7 | ss2abi 3954 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
9 | | ovex 7197 |
. . . . . . . 8
⊢
(1...𝑁) ∈
V |
10 | 9, 9 | mapval 8442 |
. . . . . . 7
⊢
((1...𝑁)
↑m (1...𝑁))
= {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
11 | 8, 10 | sseqtrri 3912 |
. . . . . 6
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁)) |
12 | | ssfi 8765 |
. . . . . 6
⊢
((((1...𝑁)
↑m (1...𝑁))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) |
13 | 6, 11, 12 | mp2an 692 |
. . . . 5
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin |
14 | 4, 13 | pm3.2i 474 |
. . . 4
⊢
(((0..^𝐾)
↑m (1...𝑁))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) |
15 | | xpfi 8856 |
. . . 4
⊢
((((0..^𝐾)
↑m (1...𝑁))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin) |
16 | 14, 15 | mp1i 13 |
. . 3
⊢ (𝜑 → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin) |
17 | | 2z 12088 |
. . . 4
⊢ 2 ∈
ℤ |
18 | 17 | a1i 11 |
. . 3
⊢ (𝜑 → 2 ∈
ℤ) |
19 | | snfi 8635 |
. . . . . . 7
⊢ {𝑥} ∈ Fin |
20 | | fzfi 13424 |
. . . . . . . 8
⊢
(0...𝑁) ∈
Fin |
21 | | rabfi 8814 |
. . . . . . . 8
⊢
((0...𝑁) ∈ Fin
→ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) |
22 | 20, 21 | ax-mp 5 |
. . . . . . 7
⊢ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin |
23 | | xpfi 8856 |
. . . . . . 7
⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin) |
24 | 19, 22, 23 | mp2an 692 |
. . . . . 6
⊢ ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin |
25 | | hashcl 13802 |
. . . . . 6
⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈
ℕ0) |
26 | 24, 25 | ax-mp 5 |
. . . . 5
⊢
(♯‘({𝑥}
× {𝑦 ∈
(0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈
ℕ0 |
27 | 26 | nn0zi 12081 |
. . . 4
⊢
(♯‘({𝑥}
× {𝑦 ∈
(0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ |
28 | 27 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ) |
29 | | poimir.0 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
30 | 29 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 𝑁 ∈ ℕ) |
31 | | nfv 1920 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) |
32 | | nfcsb1v 3812 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 |
33 | 32 | nfeq2 2916 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 |
34 | 31, 33 | nfim 1902 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
35 | | oveq2 7172 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘)) |
36 | 35 | imaeq2d 5897 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ (1...𝑗)) = ((2nd
‘𝑡) “
(1...𝑘))) |
37 | 36 | xpeq1d 5548 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑡)
“ (1...𝑘)) ×
{1})) |
38 | | oveq1 7171 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1)) |
39 | 38 | oveq1d 7179 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑘 → ((𝑗 + 1)...𝑁) = ((𝑘 + 1)...𝑁)) |
40 | 39 | imaeq2d 5897 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁))) |
41 | 40 | xpeq1d 5548 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})) |
42 | 37, 41 | uneq12d 4052 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑁)) × {0})) =
((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) |
43 | 42 | oveq2d 7180 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘𝑡)
∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))) |
44 | 43 | eqeq2d 2749 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd
‘𝑡) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))))) |
45 | | csbeq1a 3802 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
46 | 45 | eqeq2d 2749 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝐵 = ⦋𝑡 / 𝑠⦌𝐶 ↔ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
47 | 44, 46 | imbi12d 348 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → ((𝑝 = ((1st ‘𝑡) ∘f + ((((2nd
‘𝑡) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd
‘𝑡) “
(1...𝑘)) × {1}) ∪
(((2nd ‘𝑡)
“ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))) |
48 | | nfv 1920 |
. . . . . . . . . . 11
⊢
Ⅎ𝑠 𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) |
49 | | nfcsb1v 3812 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑠⦋𝑡 / 𝑠⦌𝐶 |
50 | 49 | nfeq2 2916 |
. . . . . . . . . . 11
⊢
Ⅎ𝑠 𝐵 = ⦋𝑡 / 𝑠⦌𝐶 |
51 | 48, 50 | nfim 1902 |
. . . . . . . . . 10
⊢
Ⅎ𝑠(𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶) |
52 | | fveq2 6668 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡)) |
53 | | fveq2 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡)) |
54 | 53 | imaeq1d 5896 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘𝑡) “
(1...𝑗))) |
55 | 54 | xpeq1d 5548 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑡)
“ (1...𝑗)) ×
{1})) |
56 | 53 | imaeq1d 5896 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁))) |
57 | 56 | xpeq1d 5548 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})) |
58 | 55, 57 | uneq12d 4052 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0})) =
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) |
59 | 52, 58 | oveq12d 7182 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘𝑡)
∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))) |
60 | 59 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
61 | | csbeq1a 3802 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → 𝐶 = ⦋𝑡 / 𝑠⦌𝐶) |
62 | 61 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → (𝐵 = 𝐶 ↔ 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)) |
63 | 60, 62 | imbi12d 348 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑡 → ((𝑝 = ((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd
‘𝑡) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶))) |
64 | | poimirlem28.1 |
. . . . . . . . . 10
⊢ (𝑝 = ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) |
65 | 51, 63, 64 | chvarfv 2241 |
. . . . . . . . 9
⊢ (𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶) |
66 | 34, 47, 65 | chvarfv 2241 |
. . . . . . . 8
⊢ (𝑝 = ((1st ‘𝑡) ∘f +
((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
67 | | poimirlem28.2 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
68 | 67 | ad4ant14 752 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
69 | | xp1st 7739 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
70 | | elmapi 8452 |
. . . . . . . . . 10
⊢
((1st ‘𝑥) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾)) |
71 | 69, 70 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾)) |
72 | 71 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾)) |
73 | | xp2nd 7740 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
74 | | fvex 6681 |
. . . . . . . . . . 11
⊢
(2nd ‘𝑥) ∈ V |
75 | | f1oeq1 6600 |
. . . . . . . . . . 11
⊢ (𝑓 = (2nd ‘𝑥) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))) |
76 | 74, 75 | elab 3571 |
. . . . . . . . . 10
⊢
((2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁)) |
77 | 73, 76 | sylib 221 |
. . . . . . . . 9
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁)) |
78 | 77 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁)) |
79 | | nfcv 2899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝑁 |
80 | | nfcv 2899 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗𝑥 |
81 | 80, 32 | nfcsbw 3814 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 |
82 | 79, 81 | nfne 3034 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 |
83 | | nfcv 2899 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡𝐶 |
84 | 83, 49, 61 | cbvcsbw 3798 |
. . . . . . . . . . . . . 14
⊢
⦋𝑥 /
𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶 |
85 | 45 | csbeq2dv 3795 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
86 | 84, 85 | syl5eq 2785 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
87 | 86 | neeq2d 2994 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
88 | 82, 87 | rspc 3512 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
89 | 88 | impcom 411 |
. . . . . . . . . 10
⊢
((∀𝑗 ∈
(0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
90 | 89 | adantll 714 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
91 | | 1st2nd2 7746 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
92 | 91 | csbeq1d 3792 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
93 | 92 | ad3antlr 731 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
94 | 90, 93 | neeqtrd 3003 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
95 | 30, 66, 68, 72, 78, 94 | poimirlem25 35414 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶})) |
96 | | nfv 1920 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘 𝑖 = ⦋𝑥 / 𝑠⦌𝐶 |
97 | 81 | nfeq2 2916 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 |
98 | 86 | eqeq2d 2749 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑘 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
99 | 96, 97, 98 | cbvrexw 3340 |
. . . . . . . . . . . . 13
⊢
(∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶) |
100 | 92 | eqeq2d 2749 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
101 | 100 | rexbidv 3206 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)) |
102 | 99, 101 | syl5rbb 287 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶)) |
103 | 102 | ralbidv 3109 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶)) |
104 | | iba 531 |
. . . . . . . . . . 11
⊢
(∀𝑗 ∈
(0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))) |
105 | 103, 104 | sylan9bb 513 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))) |
106 | 105 | rabbidv 3380 |
. . . . . . . . 9
⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) |
107 | 106 | fveq2d 6672 |
. . . . . . . 8
⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
108 | 107 | adantll 714 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋〈(1st
‘𝑥), (2nd
‘𝑥)〉 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
109 | 95, 108 | breqtrd 5053 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
110 | 109 | ex 416 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
111 | | dvds0 15710 |
. . . . . . . 8
⊢ (2 ∈
ℤ → 2 ∥ 0) |
112 | 17, 111 | ax-mp 5 |
. . . . . . 7
⊢ 2 ∥
0 |
113 | | hash0 13813 |
. . . . . . 7
⊢
(♯‘∅) = 0 |
114 | 112, 113 | breqtrri 5054 |
. . . . . 6
⊢ 2 ∥
(♯‘∅) |
115 | | simpr 488 |
. . . . . . . . . 10
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) |
116 | 115 | con3i 157 |
. . . . . . . . 9
⊢ (¬
∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) |
117 | 116 | ralrimivw 3097 |
. . . . . . . 8
⊢ (¬
∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) |
118 | | rabeq0 4270 |
. . . . . . . 8
⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅ ↔ ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) |
119 | 117, 118 | sylibr 237 |
. . . . . . 7
⊢ (¬
∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅) |
120 | 119 | fveq2d 6672 |
. . . . . 6
⊢ (¬
∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) =
(♯‘∅)) |
121 | 114, 120 | breqtrrid 5065 |
. . . . 5
⊢ (¬
∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
122 | 110, 121 | pm2.61d1 183 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
123 | | hashxp 13880 |
. . . . . 6
⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((♯‘{𝑥}) · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
124 | 19, 22, 123 | mp2an 692 |
. . . . 5
⊢
(♯‘({𝑥}
× {𝑦 ∈
(0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((♯‘{𝑥}) · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
125 | | vex 3401 |
. . . . . . 7
⊢ 𝑥 ∈ V |
126 | | hashsng 13815 |
. . . . . . 7
⊢ (𝑥 ∈ V →
(♯‘{𝑥}) =
1) |
127 | 125, 126 | ax-mp 5 |
. . . . . 6
⊢
(♯‘{𝑥})
= 1 |
128 | 127 | oveq1i 7174 |
. . . . 5
⊢
((♯‘{𝑥})
· (♯‘{𝑦
∈ (0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (1 · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
129 | | hashcl 13802 |
. . . . . . . 8
⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin → (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈
ℕ0) |
130 | 22, 129 | ax-mp 5 |
. . . . . . 7
⊢
(♯‘{𝑦
∈ (0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈
ℕ0 |
131 | 130 | nn0cni 11981 |
. . . . . 6
⊢
(♯‘{𝑦
∈ (0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℂ |
132 | 131 | mulid2i 10717 |
. . . . 5
⊢ (1
· (♯‘{𝑦
∈ (0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) |
133 | 124, 128,
132 | 3eqtri 2765 |
. . . 4
⊢
(♯‘({𝑥}
× {𝑦 ∈
(0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) |
134 | 122, 133 | breqtrrdi 5069 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
135 | 16, 18, 28, 134 | fsumdvds 15746 |
. 2
⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
136 | 4, 13, 15 | mp2an 692 |
. . . . . 6
⊢
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin |
137 | | xpfi 8856 |
. . . . . 6
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin) |
138 | 136, 20, 137 | mp2an 692 |
. . . . 5
⊢
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin |
139 | | rabfi 8814 |
. . . . 5
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin) |
140 | 138, 139 | ax-mp 5 |
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin |
141 | 29 | nncnd 11725 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℂ) |
142 | | npcan1 11136 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
143 | 141, 142 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
144 | | nnm1nn0 12010 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
145 | 29, 144 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
146 | 145 | nn0zd 12159 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
147 | | uzid 12332 |
. . . . . . . . . . . 12
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
148 | | peano2uz 12376 |
. . . . . . . . . . . 12
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
149 | 146, 147,
148 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
150 | 143, 149 | eqeltrrd 2834 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
151 | | fzss2 13031 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁)) |
152 | | ssralv 3941 |
. . . . . . . . . 10
⊢
((0...(𝑁 − 1))
⊆ (0...𝑁) →
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
153 | 150, 151,
152 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
154 | 153 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
155 | | raldifb 4033 |
. . . . . . . . . . . 12
⊢
(∀𝑗 ∈
(0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
156 | | nfv 1920 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜑 |
157 | | nfcsb1v 3812 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
158 | 157 | nfeq2 2916 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑁 =
⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
159 | 156, 158 | nfan 1905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
160 | | nfv 1920 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗 𝑖 ∈ (0...(𝑁 − 1)) |
161 | 159, 160 | nfan 1905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) |
162 | | nnel 3047 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑗 ∉ {(2nd
‘𝑡)} ↔ 𝑗 ∈ {(2nd
‘𝑡)}) |
163 | | velsn 4529 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ {(2nd
‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡)) |
164 | 162, 163 | bitri 278 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑗 ∉ {(2nd
‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡)) |
165 | | csbeq1a 3802 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (2nd ‘𝑡) →
⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
166 | 165 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (2nd ‘𝑡) → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
167 | 166 | biimparc 483 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 =
⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
168 | 29 | nnred 11724 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ ℝ) |
169 | 168 | ltm1d 11643 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
170 | 145 | nn0red 12030 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
171 | 170, 168 | ltnled 10858 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1))) |
172 | 169, 171 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1)) |
173 | | elfzle2 12995 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
174 | 172, 173 | nsyl 142 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1))) |
175 | | eleq1 2820 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 = 𝑁 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ 𝑁 ∈ (0...(𝑁 − 1)))) |
176 | 175 | notbid 321 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = 𝑁 → (¬ 𝑖 ∈ (0...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))) |
177 | 174, 176 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑖 = 𝑁 → ¬ 𝑖 ∈ (0...(𝑁 − 1)))) |
178 | 177 | con2d 136 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑖 ∈ (0...(𝑁 − 1)) → ¬ 𝑖 = 𝑁)) |
179 | 178 | imp 410 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ¬ 𝑖 = 𝑁) |
180 | | eqeq2 2750 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 = 𝑁 ↔ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
181 | 180 | notbid 321 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑖 = 𝑁 ↔ ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
182 | 179, 181 | syl5ibcom 248 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
183 | 167, 182 | syl5 34 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
184 | 183 | expdimp 456 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
185 | 184 | an32s 652 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
186 | 164, 185 | syl5bi 245 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑗 ∉ {(2nd
‘𝑡)} → ¬
𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶)) |
187 | | idd 24 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
188 | 186, 187 | jad 190 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
189 | 188 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
190 | 161, 189 | ralimdaa 3128 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
191 | 155, 190 | syl5bir 246 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
192 | 191 | con3d 155 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
193 | | dfrex2 3151 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
(0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
194 | | dfrex2 3151 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
195 | 192, 193,
194 | 3imtr4g 299 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
196 | 195 | ralimdva 3091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
197 | 154, 196 | syld 47 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
198 | 197 | expimpd 457 |
. . . . . 6
⊢ (𝜑 → ((𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
199 | 198 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
200 | 199 | ss2rabdv 3963 |
. . . 4
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) |
201 | | hashssdif 13858 |
. . . 4
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}))) |
202 | 140, 200,
201 | sylancr 590 |
. . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}))) |
203 | | xp2nd 7740 |
. . . . . . . 8
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁)) |
204 | | df-ne 2935 |
. . . . . . . . . . . 12
⊢ (𝑁 ≠
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
205 | 204 | ralbii 3080 |
. . . . . . . . . . 11
⊢
(∀𝑗 ∈
(0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
206 | | ralnex 3148 |
. . . . . . . . . . 11
⊢
(∀𝑗 ∈
(0...𝑁) ¬ 𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
207 | 205, 206 | bitri 278 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
208 | 29 | nnnn0d 12029 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
209 | | nn0uz 12355 |
. . . . . . . . . . . . . . . . . . 19
⊢
ℕ0 = (ℤ≥‘0) |
210 | 208, 209 | eleqtrdi 2843 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
211 | 143, 210 | eqeltrd 2833 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘0)) |
212 | | fzsplit2 13016 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
213 | 211, 150,
212 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
214 | 143 | oveq1d 7179 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁)) |
215 | 29 | nnzd 12160 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℤ) |
216 | | fzsn 13033 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) |
217 | 215, 216 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
218 | 214, 217 | eqtrd 2773 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁}) |
219 | 218 | uneq2d 4051 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁})) |
220 | 213, 219 | eqtrd 2773 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁})) |
221 | 220 | raleqdv 3315 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
222 | | ralunb 4079 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑖 ∈
((0...(𝑁 − 1)) ∪
{𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
223 | | difss 4020 |
. . . . . . . . . . . . . . . . . 18
⊢
((0...𝑁) ∖
{(2nd ‘𝑡)}) ⊆ (0...𝑁) |
224 | | ssrexv 3942 |
. . . . . . . . . . . . . . . . . 18
⊢
(((0...𝑁) ∖
{(2nd ‘𝑡)}) ⊆ (0...𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
225 | 223, 224 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
226 | 225 | ralimi 3075 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
227 | 226 | biantrurd 536 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
228 | 222, 227 | bitr4id 293 |
. . . . . . . . . . . . . 14
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
229 | 221, 228 | sylan9bb 513 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
230 | 229 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
231 | | nn0fz0 13089 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ0
↔ 𝑁 ∈ (0...𝑁)) |
232 | 208, 231 | sylib 221 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
233 | 232 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 ∈ (0...𝑁)) |
234 | | eqeq1 2742 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
235 | 234 | rexbidv 3206 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑁 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
236 | 235 | rspcva 3522 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
237 | | nfv 1920 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗(𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) |
238 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑗(0...(𝑁 − 1)) |
239 | | nfre1 3215 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑗∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
240 | 238, 239 | nfralw 3137 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
241 | 237, 240 | nfan 1905 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
242 | | eleq1 2820 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑁 ∈ (0...(𝑁 − 1)) ↔
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
243 | 242 | notbid 321 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ ¬
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
244 | 174, 243 | syl5ibcom 248 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
245 | 244 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
246 | | eldifsn 4672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ↔ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡))) |
247 | | diffi 8820 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((0...𝑁) ∈ Fin
→ ((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin) |
248 | 20, 247 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin |
249 | | ssrab2 3967 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd
‘𝑡)}) |
250 | | ssdomg 8594 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin → ({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd
‘𝑡)}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd
‘𝑡)}))) |
251 | 248, 249,
250 | mp2 9 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd
‘𝑡)}) |
252 | | hashdifsn 13860 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((0...𝑁) ∈ Fin
∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {(2nd
‘𝑡)})) =
((♯‘(0...𝑁))
− 1)) |
253 | 20, 252 | mpan 690 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((2nd ‘𝑡) ∈ (0...𝑁) → (♯‘((0...𝑁) ∖ {(2nd
‘𝑡)})) =
((♯‘(0...𝑁))
− 1)) |
254 | | 1cnd 10707 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → 1 ∈
ℂ) |
255 | 141, 254,
254 | addsubd 11089 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1)) |
256 | | hashfz0 13878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑁 ∈ ℕ0
→ (♯‘(0...𝑁)) = (𝑁 + 1)) |
257 | 208, 256 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1)) |
258 | 257 | oveq1d 7179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = ((𝑁 + 1) −
1)) |
259 | | hashfz0 13878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑁 − 1) ∈
ℕ0 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1)) |
260 | 145, 259 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) +
1)) |
261 | 255, 258,
260 | 3eqtr4d 2783 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) =
(♯‘(0...(𝑁
− 1)))) |
262 | 253, 261 | sylan9eqr 2795 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) →
(♯‘((0...𝑁)
∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1)))) |
263 | | fzfi 13424 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(0...(𝑁 − 1))
∈ Fin |
264 | | hashen 13792 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) →
((♯‘((0...𝑁)
∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))) |
265 | 248, 263,
264 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
((♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) =
(♯‘(0...(𝑁
− 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1))) |
266 | 262, 265 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) → ((0...𝑁) ∖ {(2nd
‘𝑡)}) ≈
(0...(𝑁 −
1))) |
267 | | rabfi 8814 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin) |
268 | 248, 267 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin |
269 | | eleq1 2820 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 ∈ (0...(𝑁 − 1)) ↔
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
270 | 269 | biimpac 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) |
271 | | rabid 3280 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
272 | 271 | simplbi2com 506 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})) |
273 | 270, 272 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})) |
274 | 273 | impancom 455 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})) |
275 | 274 | ancrd 555 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
276 | 275 | expimpd 457 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑖 ∈ (0...(𝑁 − 1)) → ((𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
277 | 276 | reximdv2 3180 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
278 | 271 | simplbi 501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) |
279 | 274 | pm4.71rd 566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
280 | | df-mpt 5108 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) = {〈𝑘, 𝑖〉 ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)} |
281 | | nfv 1920 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
Ⅎ𝑘(𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
282 | | nfrab1 3286 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢
Ⅎ𝑗{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} |
283 | 282 | nfcri 2886 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢
Ⅎ𝑗 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} |
284 | | nfcsb1v 3812 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢
Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
285 | 284 | nfeq2 2916 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢
Ⅎ𝑗 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
286 | 283, 285 | nfan 1905 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
287 | | eleq1 2820 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢ (𝑗 = 𝑘 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})) |
288 | | csbeq1a 3802 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢ (𝑗 = 𝑘 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
289 | 288 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢ (𝑗 = 𝑘 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
290 | 287, 289 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (𝑗 = 𝑘 → ((𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
291 | 281, 286,
290 | cbvopab1 5100 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
{〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} = {〈𝑘, 𝑖〉 ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)} |
292 | 280, 291 | eqtr4i 2764 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) = {〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} |
293 | 292 | breqi 5033 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑗{〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}𝑖) |
294 | | df-br 5028 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑗{〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}𝑖 ↔ 〈𝑗, 𝑖〉 ∈ {〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) |
295 | | opabidw 5377 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(〈𝑗, 𝑖〉 ∈ {〈𝑗, 𝑖〉 ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
296 | 293, 294,
295 | 3bitri 300 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
297 | 279, 296 | bitr4di 292 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
298 | 278, 297 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
299 | 298 | rexbidva 3205 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
300 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
Ⅎ𝑝{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} |
301 | | nfv 1920 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
Ⅎ𝑝 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 |
302 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
Ⅎ𝑗𝑝 |
303 | 282, 284 | nfmpt 5124 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
304 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
Ⅎ𝑗𝑖 |
305 | 302, 303,
304 | nfbr 5074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
Ⅎ𝑗 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 |
306 | | breq1 5030 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑗 = 𝑝 → (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
307 | 282, 300,
301, 305, 306 | cbvrexfw 3336 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(∃𝑗 ∈
{𝑗 ∈ ((0...𝑁) ∖ {(2nd
‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖) |
308 | 299, 307 | bitrdi 290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
309 | 277, 308 | sylibd 242 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
310 | 309 | ralimia 3073 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖) |
311 | | eqid 2738 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) = (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
312 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
Ⅎ𝑗𝑘 |
313 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
Ⅎ𝑗((0...𝑁) ∖ {(2nd ‘𝑡)}) |
314 | 284 | nfel1 2915 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) |
315 | 288 | eleq1d 2817 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑗 = 𝑘 → (⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
316 | 312, 313,
314, 315 | elrabf 3581 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑘 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
317 | 316 | simprbi 500 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) |
318 | 311, 317 | fmpti 6880 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) |
319 | 310, 318 | jctil 523 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
320 | | dffo4 6873 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)) ↔ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)𝑖)) |
321 | 319, 320 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1))) |
322 | | fodomfi 8863 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin ∧ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1))) → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) |
323 | 268, 321,
322 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) |
324 | | endomtr 8606 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((((0...𝑁) ∖
{(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)) ∧ (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → ((0...𝑁) ∖ {(2nd
‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) |
325 | 266, 323,
324 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) |
326 | | sbth 8680 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd
‘𝑡)}) ∧
((0...𝑁) ∖
{(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd
‘𝑡)})) |
327 | 251, 325,
326 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd
‘𝑡)})) |
328 | | fisseneq 8801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((0...𝑁) ∖
{(2nd ‘𝑡)}) ∈ Fin ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd
‘𝑡)}) ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd
‘𝑡)})) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)})) |
329 | 248, 249,
327, 328 | mp3an12i 1466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)})) |
330 | 329 | eleq2d 2818 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))) |
331 | 330 | biimpar 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) |
332 | 288 | equcoms 2031 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑗 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
333 | 332 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
334 | 333 | eleq1d 2817 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
335 | 334, 317 | vtoclga 3477 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} →
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) |
336 | 331, 335 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) →
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) |
337 | 246, 336 | sylan2br 598 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡))) →
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))) |
338 | 337 | expr 460 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑗 ≠ (2nd ‘𝑡) →
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))) |
339 | 338 | necon1bd 2952 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (¬
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → 𝑗 = (2nd ‘𝑡))) |
340 | 245, 339 | syld 47 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑗 = (2nd ‘𝑡))) |
341 | 340 | imp 410 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧
(2nd ‘𝑡)
∈ (0...𝑁)) ∧
∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑗 = (2nd ‘𝑡)) |
342 | 341, 165 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧
(2nd ‘𝑡)
∈ (0...𝑁)) ∧
∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
343 | | eqtr 2758 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
344 | 343 | ex 416 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
345 | 344 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧
(2nd ‘𝑡)
∈ (0...𝑁)) ∧
∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
346 | 342, 345 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧
(2nd ‘𝑡)
∈ (0...𝑁)) ∧
∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖
{(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
347 | 346 | exp31 423 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ (0...𝑁) → (𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
348 | 241, 158,
347 | rexlimd 3226 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
349 | 236, 348 | syl5 34 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
350 | 233, 349 | mpand 695 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
351 | 350 | pm4.71rd 566 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
352 | 235 | ralsng 4561 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ →
(∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
353 | 29, 352 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
354 | 353 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
355 | 230, 351,
354 | 3bitr3rd 313 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
356 | 355 | notbid 321 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
357 | 207, 356 | syl5bb 286 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
358 | 357 | pm5.32da 582 |
. . . . . . . 8
⊢ ((𝜑 ∧ (2nd
‘𝑡) ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)))) |
359 | 203, 358 | sylan2 596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)))) |
360 | 359 | rabbidva 3378 |
. . . . . 6
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))}) |
361 | | nfv 1920 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑡 = 〈𝑥, 𝑘〉 |
362 | | nfv 1920 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
363 | | nfrab1 3286 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} |
364 | 363 | nfcri 2886 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} |
365 | 362, 364 | nfan 1905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦(𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) |
366 | 361, 365 | nfan 1905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑡 = 〈𝑥, 𝑘〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
367 | | nfv 1920 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘(𝑡 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))) |
368 | | opeq2 4758 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑦 → 〈𝑥, 𝑘〉 = 〈𝑥, 𝑦〉) |
369 | 368 | eqeq2d 2749 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → (𝑡 = 〈𝑥, 𝑘〉 ↔ 𝑡 = 〈𝑥, 𝑦〉)) |
370 | | eleq1 2820 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ 𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
371 | | rabid 3280 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))) |
372 | 370, 371 | bitrdi 290 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
373 | 372 | anbi2d 632 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))) |
374 | | 3anass 1096 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
375 | 373, 374 | bitr4di 292 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
376 | 369, 375 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑦 → ((𝑡 = 〈𝑥, 𝑘〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ (𝑡 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))) |
377 | 366, 367,
376 | cbvexv1 2343 |
. . . . . . . . . 10
⊢
(∃𝑘(𝑡 = 〈𝑥, 𝑘〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
378 | 377 | exbii 1854 |
. . . . . . . . 9
⊢
(∃𝑥∃𝑘(𝑡 = 〈𝑥, 𝑘〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑥∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
379 | | eliunxp 5674 |
. . . . . . . . 9
⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∃𝑥∃𝑘(𝑡 = 〈𝑥, 𝑘〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
380 | | elopab 5379 |
. . . . . . . . 9
⊢ (𝑡 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))} ↔ ∃𝑥∃𝑦(𝑡 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))) |
381 | 378, 379,
380 | 3bitr4i 306 |
. . . . . . . 8
⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ 𝑡 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))}) |
382 | 381 | eqriv 2735 |
. . . . . . 7
⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))} |
383 | | vex 3401 |
. . . . . . . . . . . . . 14
⊢ 𝑦 ∈ V |
384 | 125, 383 | op2ndd 7718 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (2nd ‘𝑡) = 𝑦) |
385 | 384 | sneqd 4525 |
. . . . . . . . . . . 12
⊢ (𝑡 = 〈𝑥, 𝑦〉 → {(2nd ‘𝑡)} = {𝑦}) |
386 | 385 | difeq2d 4011 |
. . . . . . . . . . 11
⊢ (𝑡 = 〈𝑥, 𝑦〉 → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑦})) |
387 | 125, 383 | op1std 7717 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (1st ‘𝑡) = 𝑥) |
388 | 387 | csbeq1d 3792 |
. . . . . . . . . . . 12
⊢ (𝑡 = 〈𝑥, 𝑦〉 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋𝑥 / 𝑠⦌𝐶) |
389 | 388 | eqeq2d 2749 |
. . . . . . . . . . 11
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑠⦌𝐶)) |
390 | 386, 389 | rexeqbidv 3304 |
. . . . . . . . . 10
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶)) |
391 | 390 | ralbidv 3109 |
. . . . . . . . 9
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶)) |
392 | 388 | neeq2d 2994 |
. . . . . . . . . 10
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) |
393 | 392 | ralbidv 3109 |
. . . . . . . . 9
⊢ (𝑡 = 〈𝑥, 𝑦〉 → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) |
394 | 391, 393 | anbi12d 634 |
. . . . . . . 8
⊢ (𝑡 = 〈𝑥, 𝑦〉 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))) |
395 | 394 | rabxp 5565 |
. . . . . . 7
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))} |
396 | 382, 395 | eqtr4i 2764 |
. . . . . 6
⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} |
397 | | difrab 4195 |
. . . . . 6
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))} |
398 | 360, 396,
397 | 3eqtr4g 2798 |
. . . . 5
⊢ (𝜑 → ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) |
399 | 398 | fveq2d 6672 |
. . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}))) |
400 | 24 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin) |
401 | | inxp 5669 |
. . . . . . . . . 10
⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) |
402 | | df-ne 2935 |
. . . . . . . . . . . . 13
⊢ (𝑥 ≠ 𝑡 ↔ ¬ 𝑥 = 𝑡) |
403 | | disjsn2 4600 |
. . . . . . . . . . . . 13
⊢ (𝑥 ≠ 𝑡 → ({𝑥} ∩ {𝑡}) = ∅) |
404 | 402, 403 | sylbir 238 |
. . . . . . . . . . . 12
⊢ (¬
𝑥 = 𝑡 → ({𝑥} ∩ {𝑡}) = ∅) |
405 | 404 | xpeq1d 5548 |
. . . . . . . . . . 11
⊢ (¬
𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}))) |
406 | | 0xp 5614 |
. . . . . . . . . . 11
⊢ (∅
× ({𝑦 ∈
(0...𝑁) ∣
(∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅ |
407 | 405, 406 | eqtrdi 2789 |
. . . . . . . . . 10
⊢ (¬
𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅) |
408 | 401, 407 | syl5eq 2785 |
. . . . . . . . 9
⊢ (¬
𝑥 = 𝑡 → (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅) |
409 | 408 | orri 861 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅) |
410 | 409 | rgen2w 3066 |
. . . . . . 7
⊢
∀𝑥 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅) |
411 | | sneq 4523 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → {𝑥} = {𝑡}) |
412 | | csbeq1 3791 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑡 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑡 / 𝑠⦌𝐶) |
413 | 412 | eqeq2d 2749 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑡 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑠⦌𝐶)) |
414 | 413 | rexbidv 3206 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶)) |
415 | 414 | ralbidv 3109 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶)) |
416 | 412 | neeq2d 2994 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑡 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)) |
417 | 416 | ralbidv 3109 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)) |
418 | 415, 417 | anbi12d 634 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑡 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))) |
419 | 418 | rabbidv 3380 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}) |
420 | 411, 419 | xpeq12d 5550 |
. . . . . . . 8
⊢ (𝑥 = 𝑡 → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) |
421 | 420 | disjor 5007 |
. . . . . . 7
⊢
(Disj 𝑥
∈ (((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∀𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)) |
422 | 410, 421 | mpbir 234 |
. . . . . 6
⊢
Disj 𝑥 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) |
423 | 422 | a1i 11 |
. . . . 5
⊢ (𝜑 → Disj 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) |
424 | 16, 400, 423 | hashiun 15263 |
. . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
425 | 399, 424 | eqtr3d 2775 |
. . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))) |
426 | | fo1st 7727 |
. . . . . . . . . . . 12
⊢
1st :V–onto→V |
427 | | fofun 6587 |
. . . . . . . . . . . 12
⊢
(1st :V–onto→V → Fun 1st ) |
428 | 426, 427 | ax-mp 5 |
. . . . . . . . . . 11
⊢ Fun
1st |
429 | | ssv 3899 |
. . . . . . . . . . . 12
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ V |
430 | | fof 6586 |
. . . . . . . . . . . . . 14
⊢
(1st :V–onto→V → 1st
:V⟶V) |
431 | 426, 430 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
1st :V⟶V |
432 | 431 | fdmi 6510 |
. . . . . . . . . . . 12
⊢ dom
1st = V |
433 | 429, 432 | sseqtrri 3912 |
. . . . . . . . . . 11
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ dom
1st |
434 | | fores 6596 |
. . . . . . . . . . 11
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) →
(1st ↾ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) |
435 | 428, 433,
434 | mp2an 692 |
. . . . . . . . . 10
⊢
(1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) |
436 | | fveq2 6668 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥)) |
437 | 436 | csbeq1d 3792 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
438 | | fveq2 6668 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥)) |
439 | 438 | csbeq1d 3792 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
440 | 439 | csbeq2dv 3795 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
441 | 437, 440 | eqtrd 2773 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
442 | 441 | eqeq2d 2749 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
443 | 439 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
444 | 443 | rexbidv 3206 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
445 | 444 | ralbidv 3109 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
446 | 442, 445 | anbi12d 634 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑥 → ((𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
447 | 446 | rexrab 3593 |
. . . . . . . . . . . . . 14
⊢
(∃𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)) |
448 | | xp1st 7739 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
449 | 448 | anim1i 618 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
450 | | eleq1 2820 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))) |
451 | | csbeq1a 3802 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
452 | 451 | eqcoms 2746 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
453 | 452 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = 𝐶) |
454 | 453 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) |
455 | 454 | rexbidv 3206 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
456 | 455 | ralbidv 3109 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
457 | 450, 456 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
458 | 449, 457 | syl5ibcom 248 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
459 | 458 | adantrl 716 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
460 | 459 | expimpd 457 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
461 | 460 | rexlimiv 3189 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
462 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
463 | | ovex 7197 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(0...𝑁) ∈
V |
464 | 463 | enref 8581 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(0...𝑁) ≈
(0...𝑁) |
465 | | phpreu 35373 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((0...𝑁) ∈ Fin
∧ (0...𝑁) ≈
(0...𝑁)) →
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
466 | 20, 464, 465 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |
467 | 466 | biimpi 219 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |
468 | | eqeq1 2742 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = 𝑁 → (𝑖 = 𝐶 ↔ 𝑁 = 𝐶)) |
469 | 468 | reubidv 3291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 = 𝑁 → (∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)) |
470 | 469 | rspcva 3522 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶) |
471 | 232, 467,
470 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶) |
472 | | riotacl 7139 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∃!𝑗 ∈
(0...𝑁)𝑁 = 𝐶 → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) |
473 | 471, 472 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) |
474 | 473 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) |
475 | | opelxpi 5556 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) → 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
476 | 462, 474,
475 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
477 | | riotasbc 7140 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃!𝑗 ∈
(0...𝑁)𝑁 = 𝐶 → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶) |
478 | 471, 477 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶) |
479 | | riotaex 7125 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(℩𝑗
∈ (0...𝑁)𝑁 = 𝐶) ∈ V |
480 | | sbceq2g 4303 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((℩𝑗
∈ (0...𝑁)𝑁 = 𝐶) ∈ V →
([(℩𝑗
∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)) |
481 | 479, 480 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
([(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶) |
482 | 478, 481 | sylib 221 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶) |
483 | 482 | expcom 417 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → (𝜑 → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)) |
484 | 483 | imdistanri 573 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
485 | 484 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
486 | | vex 3401 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝑠 ∈ V |
487 | 486, 479 | op2ndd 7718 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (2nd ‘𝑥) = (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)) |
488 | 487 | csbeq1d 3792 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 →
⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶) |
489 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑗𝑠 |
490 | | nfriota1 7128 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑗(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) |
491 | 489, 490 | nfop 4774 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑗〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 |
492 | 491 | nfeq2 2916 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑗 𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 |
493 | 486, 479 | op1std 7717 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (1st ‘𝑥) = 𝑠) |
494 | 493 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → 𝑠 = (1st ‘𝑥)) |
495 | 494, 451 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
496 | 492, 495 | csbeq2d 3794 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 →
⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
497 | 488, 496 | eqtr3d 2775 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 →
⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
498 | 497 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ↔ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
499 | 495 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (𝑖 = 𝐶 ↔ 𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
500 | 492, 499 | rexbid 3229 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
501 | 500 | ralbidv 3109 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
502 | 498, 501 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → ((𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) ↔ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
503 | 493 | biantrud 535 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → ((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ↔ ((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))) |
504 | 502, 503 | bitr2d 283 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 〈𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)〉 → (((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
505 | 504 | rspcev 3524 |
. . . . . . . . . . . . . . . . 17
⊢
((〈𝑠,
(℩𝑗 ∈
(0...𝑁)𝑁 = 𝐶)〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)) |
506 | 476, 485,
505 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)) |
507 | 506 | expl 461 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))) |
508 | 461, 507 | impbid2 229 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
509 | 447, 508 | syl5bb 286 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))) |
510 | 509 | abbidv 2802 |
. . . . . . . . . . . 12
⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)}) |
511 | | dfimafn 6726 |
. . . . . . . . . . . . . 14
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) →
(1st “ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦}) |
512 | 428, 433,
511 | mp2an 692 |
. . . . . . . . . . . . 13
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦} |
513 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠(2nd ‘𝑡) |
514 | | nfcsb1v 3812 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
515 | 513, 514 | nfcsbw 3814 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
516 | 515 | nfeq2 2916 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠 𝑁 =
⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
517 | | nfcv 2899 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(0...𝑁) |
518 | 514 | nfeq2 2916 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
519 | 517, 518 | nfrex 3218 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
520 | 517, 519 | nfralw 3137 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
521 | 516, 520 | nfan 1905 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠(𝑁 =
⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
522 | | nfcv 2899 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) |
523 | 521, 522 | nfrabw 3287 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} |
524 | | nfv 1920 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠(1st ‘𝑥) = 𝑦 |
525 | 523, 524 | nfrex 3218 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦 |
526 | | nfv 1920 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 |
527 | | eqeq2 2750 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠)) |
528 | 527 | rexbidv 3206 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠)) |
529 | 525, 526,
528 | cbvabw 2807 |
. . . . . . . . . . . . 13
⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠} |
530 | 512, 529 | eqtri 2761 |
. . . . . . . . . . . 12
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠} |
531 | | df-rab 3062 |
. . . . . . . . . . . 12
⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)} |
532 | 510, 530,
531 | 3eqtr4g 2798 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st “
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
533 | | foeq3 6584 |
. . . . . . . . . . 11
⊢
((1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
534 | 532, 533 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
535 | 435, 534 | mpbii 236 |
. . . . . . . . 9
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
536 | | fof 6586 |
. . . . . . . . 9
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
537 | 535, 536 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
538 | | fvres 6687 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = (1st ‘𝑥)) |
539 | | fvres 6687 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑦) = (1st ‘𝑦)) |
540 | 538, 539 | eqeqan12d 2755 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦))) |
541 | 540 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦))) |
542 | 446 | elrab 3585 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
543 | | xp2nd 7740 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑥) ∈ (0...𝑁)) |
544 | 543 | anim1i 618 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
545 | 542, 544 | sylbi 220 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
546 | | simpl 486 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 =
⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
547 | 546 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
548 | 547 | ss2rabi 3964 |
. . . . . . . . . . . . . . . 16
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶} |
549 | 548 | sseli 3871 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶}) |
550 | | fveq2 6668 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦)) |
551 | 550 | csbeq1d 3792 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑦 → ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
552 | | fveq2 6668 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑦 → (1st ‘𝑡) = (1st ‘𝑦)) |
553 | 552 | csbeq1d 3792 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑦 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑦) / 𝑠⦌𝐶) |
554 | 553 | csbeq2dv 3795 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑦 → ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶) |
555 | 551, 554 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑦 → ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶) |
556 | 555 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑦 → (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) |
557 | 556 | elrab 3585 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶} ↔ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) |
558 | | xp2nd 7740 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑦) ∈ (0...𝑁)) |
559 | 558 | anim1i 618 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶) → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) |
560 | 557, 559 | sylbi 220 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) |
561 | 549, 560 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) |
562 | 545, 561 | anim12i 616 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) → (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) |
563 | | an4 656 |
. . . . . . . . . . . . . . 15
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) |
564 | 563 | anbi2i 626 |
. . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)))) |
565 | | anass 472 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ↔ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
566 | | ancom 464 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
567 | 565, 566 | bitr3i 280 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶))) |
568 | 567 | anbi1i 627 |
. . . . . . . . . . . . . . 15
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) |
569 | | anass 472 |
. . . . . . . . . . . . . . 15
⊢
(((∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)))) |
570 | 568, 569 | bitri 278 |
. . . . . . . . . . . . . 14
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)))) |
571 | | anass 472 |
. . . . . . . . . . . . . 14
⊢
(((∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)))) |
572 | 564, 570,
571 | 3bitr4i 306 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) |
573 | 562, 572 | sylib 221 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd
‘𝑡) / 𝑗⦌⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)}) → ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd
‘𝑥) / 𝑗⦌⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd
‘𝑦) / 𝑗⦌⦋(1st
‘𝑦) / 𝑠⦌𝐶))) |
574 | | phpreu 35373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((0...𝑁) ∈ Fin
∧ (0...𝑁) ≈
(0...𝑁)) →
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
575 | 20, 464, 574 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑖 ∈
(0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
576 | | reurmo 3330 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃!𝑗 ∈
(0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 → ∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
577 | 576 | ralimi 3075 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑖 ∈
(0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
578 | 575, 577 | sylbi |