Step | Hyp | Ref
| Expression |
1 | | subfacp1lem.a |
. . . . . . . 8
⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
2 | | fzfi 13330 |
. . . . . . . . 9
⊢
(1...(𝑁 + 1)) ∈
Fin |
3 | | deranglem 32311 |
. . . . . . . . 9
⊢
((1...(𝑁 + 1))
∈ Fin → {𝑓
∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin) |
4 | 2, 3 | ax-mp 5 |
. . . . . . . 8
⊢ {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin |
5 | 1, 4 | eqeltri 2909 |
. . . . . . 7
⊢ 𝐴 ∈ Fin |
6 | | subfacp1lem3.b |
. . . . . . . 8
⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)} |
7 | | ssrab2 4055 |
. . . . . . . 8
⊢ {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)} ⊆ 𝐴 |
8 | 6, 7 | eqsstri 4000 |
. . . . . . 7
⊢ 𝐵 ⊆ 𝐴 |
9 | | ssfi 8727 |
. . . . . . 7
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
10 | 5, 8, 9 | mp2an 688 |
. . . . . 6
⊢ 𝐵 ∈ Fin |
11 | 10 | elexi 3514 |
. . . . 5
⊢ 𝐵 ∈ V |
12 | 11 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ V) |
13 | | subfacp1lem3.c |
. . . . . . 7
⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} |
14 | | subfacp1lem1.k |
. . . . . . . . 9
⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) |
15 | | fzfi 13330 |
. . . . . . . . . 10
⊢
(2...(𝑁 + 1)) ∈
Fin |
16 | | diffi 8739 |
. . . . . . . . . 10
⊢
((2...(𝑁 + 1))
∈ Fin → ((2...(𝑁
+ 1)) ∖ {𝑀}) ∈
Fin) |
17 | 15, 16 | ax-mp 5 |
. . . . . . . . 9
⊢
((2...(𝑁 + 1))
∖ {𝑀}) ∈
Fin |
18 | 14, 17 | eqeltri 2909 |
. . . . . . . 8
⊢ 𝐾 ∈ Fin |
19 | | deranglem 32311 |
. . . . . . . 8
⊢ (𝐾 ∈ Fin → {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin) |
20 | 18, 19 | ax-mp 5 |
. . . . . . 7
⊢ {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin |
21 | 13, 20 | eqeltri 2909 |
. . . . . 6
⊢ 𝐶 ∈ Fin |
22 | 21 | elexi 3514 |
. . . . 5
⊢ 𝐶 ∈ V |
23 | 22 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ V) |
24 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵) |
25 | | fveq1 6663 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = 𝑏 → (𝑔‘1) = (𝑏‘1)) |
26 | 25 | eqeq1d 2823 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑏 → ((𝑔‘1) = 𝑀 ↔ (𝑏‘1) = 𝑀)) |
27 | | fveq1 6663 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = 𝑏 → (𝑔‘𝑀) = (𝑏‘𝑀)) |
28 | 27 | eqeq1d 2823 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑏 → ((𝑔‘𝑀) = 1 ↔ (𝑏‘𝑀) = 1)) |
29 | 26, 28 | anbi12d 630 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑏 → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
30 | 29, 6 | elrab2 3682 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
31 | 24, 30 | sylib 219 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
32 | 31 | simpld 495 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴) |
33 | | vex 3498 |
. . . . . . . . . . . 12
⊢ 𝑏 ∈ V |
34 | | f1oeq1 6598 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑏 → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) |
35 | | fveq1 6663 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑏 → (𝑓‘𝑦) = (𝑏‘𝑦)) |
36 | 35 | neeq1d 3075 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑏 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦)) |
37 | 36 | ralbidv 3197 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑏 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
38 | 34, 37 | anbi12d 630 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑏 → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦))) |
39 | 33, 38, 1 | elab2 3669 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ 𝐴 ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
40 | 32, 39 | sylib 219 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
41 | 40 | simpld 495 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
42 | | f1of1 6608 |
. . . . . . . . 9
⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1))) |
43 | | df-f1 6354 |
. . . . . . . . . 10
⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) ↔ (𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ Fun ◡𝑏)) |
44 | 43 | simprbi 497 |
. . . . . . . . 9
⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) → Fun ◡𝑏) |
45 | 41, 42, 44 | 3syl 18 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Fun ◡𝑏) |
46 | | f1ofn 6610 |
. . . . . . . . . . . 12
⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏 Fn (1...(𝑁 + 1))) |
47 | 41, 46 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 Fn (1...(𝑁 + 1))) |
48 | | fnresdm 6460 |
. . . . . . . . . . 11
⊢ (𝑏 Fn (1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))) = 𝑏) |
49 | | f1oeq1 6598 |
. . . . . . . . . . 11
⊢ ((𝑏 ↾ (1...(𝑁 + 1))) = 𝑏 → ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) |
50 | 47, 48, 49 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) |
51 | 41, 50 | mpbird 258 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
52 | | f1ofo 6616 |
. . . . . . . . 9
⊢ ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1))) |
53 | 51, 52 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1))) |
54 | | ssun2 4148 |
. . . . . . . . . . . . 13
⊢ {1, 𝑀} ⊆ (𝐾 ∪ {1, 𝑀}) |
55 | | derang.d |
. . . . . . . . . . . . . . 15
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
56 | | subfac.n |
. . . . . . . . . . . . . . 15
⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
57 | | subfacp1lem1.n |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℕ) |
58 | | subfacp1lem1.m |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) |
59 | | subfacp1lem1.x |
. . . . . . . . . . . . . . 15
⊢ 𝑀 ∈ V |
60 | 55, 56, 1, 57, 58, 59, 14 | subfacp1lem1 32324 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1))) |
61 | 60 | simp2d 1135 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
62 | 54, 61 | sseqtrid 4018 |
. . . . . . . . . . . 12
⊢ (𝜑 → {1, 𝑀} ⊆ (1...(𝑁 + 1))) |
63 | 62 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → {1, 𝑀} ⊆ (1...(𝑁 + 1))) |
64 | | fnssres 6464 |
. . . . . . . . . . 11
⊢ ((𝑏 Fn (1...(𝑁 + 1)) ∧ {1, 𝑀} ⊆ (1...(𝑁 + 1))) → (𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀}) |
65 | 47, 63, 64 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀}) |
66 | 31 | simprd 496 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)) |
67 | 66 | simpld 495 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) = 𝑀) |
68 | 59 | prid2 4693 |
. . . . . . . . . . . . 13
⊢ 𝑀 ∈ {1, 𝑀} |
69 | 67, 68 | syl6eqel 2921 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) ∈ {1, 𝑀}) |
70 | 66 | simprd 496 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) = 1) |
71 | | 1ex 10626 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
72 | 71 | prid1 4692 |
. . . . . . . . . . . . 13
⊢ 1 ∈
{1, 𝑀} |
73 | 70, 72 | syl6eqel 2921 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ∈ {1, 𝑀}) |
74 | | fveq2 6664 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 1 → (𝑏‘𝑥) = (𝑏‘1)) |
75 | 74 | eleq1d 2897 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 1 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘1) ∈ {1, 𝑀})) |
76 | | fveq2 6664 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑀 → (𝑏‘𝑥) = (𝑏‘𝑀)) |
77 | 76 | eleq1d 2897 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑀 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑀) ∈ {1, 𝑀})) |
78 | 71, 59, 75, 77 | ralpr 4630 |
. . . . . . . . . . . 12
⊢
(∀𝑥 ∈
{1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀} ↔ ((𝑏‘1) ∈ {1, 𝑀} ∧ (𝑏‘𝑀) ∈ {1, 𝑀})) |
79 | 69, 73, 78 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀}) |
80 | | fvres 6683 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑥) = (𝑏‘𝑥)) |
81 | 80 | eleq1d 2897 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑥) ∈ {1, 𝑀})) |
82 | 81 | ralbiia 3164 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
{1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀}) |
83 | 79, 82 | sylibr 235 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀}) |
84 | | ffnfv 6875 |
. . . . . . . . . 10
⊢ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀} ↔ ((𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀} ∧ ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀})) |
85 | 65, 83, 84 | sylanbrc 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀}) |
86 | | fveqeq2 6673 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = 1 ↔ (𝑏‘𝑀) = 1)) |
87 | 86 | rspcev 3622 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ {1, 𝑀} ∧ (𝑏‘𝑀) = 1) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1) |
88 | 68, 70, 87 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1) |
89 | | fveqeq2 6673 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 1 → ((𝑏‘𝑦) = 𝑀 ↔ (𝑏‘1) = 𝑀)) |
90 | 89 | rspcev 3622 |
. . . . . . . . . . . 12
⊢ ((1
∈ {1, 𝑀} ∧ (𝑏‘1) = 𝑀) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀) |
91 | 72, 67, 90 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀) |
92 | | eqeq2 2833 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 1 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 1)) |
93 | 92 | rexbidv 3297 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)) |
94 | | eqeq2 2833 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑀 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑀)) |
95 | 94 | rexbidv 3297 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑀 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)) |
96 | 71, 59, 93, 95 | ralpr 4630 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
{1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1 ∧ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)) |
97 | 88, 91, 96 | sylanbrc 583 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
98 | | eqcom 2828 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥) |
99 | | fvres 6683 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑦) = (𝑏‘𝑦)) |
100 | 99 | eqeq1d 2823 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑥)) |
101 | 98, 100 | syl5bb 284 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {1, 𝑀} → (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ (𝑏‘𝑦) = 𝑥)) |
102 | 101 | rexbiia 3246 |
. . . . . . . . . . 11
⊢
(∃𝑦 ∈ {1,
𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
103 | 102 | ralbii 3165 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
{1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
104 | 97, 103 | sylibr 235 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦)) |
105 | | dffo3 6861 |
. . . . . . . . 9
⊢ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀} ↔ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀} ∧ ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦))) |
106 | 85, 104, 105 | sylanbrc 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀}) |
107 | | resdif 6629 |
. . . . . . . 8
⊢ ((Fun
◡𝑏 ∧ (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1)) ∧ (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀}) → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀})) |
108 | 45, 53, 106, 107 | syl3anc 1363 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀})) |
109 | | uncom 4128 |
. . . . . . . . . . 11
⊢ ({1,
𝑀} ∪ 𝐾) = (𝐾 ∪ {1, 𝑀}) |
110 | 109, 61 | syl5eq 2868 |
. . . . . . . . . 10
⊢ (𝜑 → ({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1))) |
111 | | incom 4177 |
. . . . . . . . . . . 12
⊢ ({1,
𝑀} ∩ 𝐾) = (𝐾 ∩ {1, 𝑀}) |
112 | 60 | simp1d 1134 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅) |
113 | 111, 112 | syl5eq 2868 |
. . . . . . . . . . 11
⊢ (𝜑 → ({1, 𝑀} ∩ 𝐾) = ∅) |
114 | | uneqdifeq 4436 |
. . . . . . . . . . 11
⊢ (({1,
𝑀} ⊆ (1...(𝑁 + 1)) ∧ ({1, 𝑀} ∩ 𝐾) = ∅) → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)) |
115 | 62, 113, 114 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)) |
116 | 110, 115 | mpbid 233 |
. . . . . . . . 9
⊢ (𝜑 → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾) |
117 | 116 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾) |
118 | | reseq2 5842 |
. . . . . . . . . 10
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})) = (𝑏 ↾ 𝐾)) |
119 | | f1oeq1 6598 |
. . . . . . . . . 10
⊢ ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})) = (𝑏 ↾ 𝐾) → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))) |
120 | 118, 119 | syl 17 |
. . . . . . . . 9
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))) |
121 | | f1oeq2 6599 |
. . . . . . . . 9
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ 𝐾):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))) |
122 | | f1oeq3 6600 |
. . . . . . . . 9
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ 𝐾):𝐾–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
123 | 120, 121,
122 | 3bitrd 306 |
. . . . . . . 8
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
124 | 117, 123 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
125 | 108, 124 | mpbid 233 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾) |
126 | | ssun1 4147 |
. . . . . . . . 9
⊢ 𝐾 ⊆ (𝐾 ∪ {1, 𝑀}) |
127 | 126, 61 | sseqtrid 4018 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ⊆ (1...(𝑁 + 1))) |
128 | 127 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐾 ⊆ (1...(𝑁 + 1))) |
129 | 40 | simprd 496 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦) |
130 | | ssralv 4032 |
. . . . . . 7
⊢ (𝐾 ⊆ (1...(𝑁 + 1)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦 → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
131 | 128, 129,
130 | sylc 65 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦) |
132 | 33 | resex 5893 |
. . . . . . 7
⊢ (𝑏 ↾ 𝐾) ∈ V |
133 | | f1oeq1 6598 |
. . . . . . . 8
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓:𝐾–1-1-onto→𝐾 ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
134 | | fveq1 6663 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)) |
135 | | fvres 6683 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝐾 → ((𝑏 ↾ 𝐾)‘𝑦) = (𝑏‘𝑦)) |
136 | 134, 135 | sylan9eq 2876 |
. . . . . . . . . 10
⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → (𝑓‘𝑦) = (𝑏‘𝑦)) |
137 | 136 | neeq1d 3075 |
. . . . . . . . 9
⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦)) |
138 | 137 | ralbidva 3196 |
. . . . . . . 8
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
139 | 133, 138 | anbi12d 630 |
. . . . . . 7
⊢ (𝑓 = (𝑏 ↾ 𝐾) → ((𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦) ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦))) |
140 | 132, 139,
13 | elab2 3669 |
. . . . . 6
⊢ ((𝑏 ↾ 𝐾) ∈ 𝐶 ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
141 | 125, 131,
140 | sylanbrc 583 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾) ∈ 𝐶) |
142 | 141 | ex 413 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ 𝐵 → (𝑏 ↾ 𝐾) ∈ 𝐶)) |
143 | 57 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑁 ∈ ℕ) |
144 | 58 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ (2...(𝑁 + 1))) |
145 | | eqid 2821 |
. . . . . . . . 9
⊢ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) |
146 | | simpr 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶) |
147 | | vex 3498 |
. . . . . . . . . . . 12
⊢ 𝑐 ∈ V |
148 | | f1oeq1 6598 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑐 → (𝑓:𝐾–1-1-onto→𝐾 ↔ 𝑐:𝐾–1-1-onto→𝐾)) |
149 | | fveq1 6663 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑐 → (𝑓‘𝑦) = (𝑐‘𝑦)) |
150 | 149 | neeq1d 3075 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑐 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑦) ≠ 𝑦)) |
151 | 150 | ralbidv 3197 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑐 → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
152 | 148, 151 | anbi12d 630 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑐 → ((𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦) ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))) |
153 | 147, 152,
13 | elab2 3669 |
. . . . . . . . . . 11
⊢ (𝑐 ∈ 𝐶 ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
154 | 146, 153 | sylib 219 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
155 | 154 | simpld 495 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐:𝐾–1-1-onto→𝐾) |
156 | 55, 56, 1, 143, 144, 59, 14, 145, 155 | subfacp1lem2a 32325 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1)) |
157 | 156 | simp1d 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
158 | 55, 56, 1, 143, 144, 59, 14, 145, 155 | subfacp1lem2b 32326 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = (𝑐‘𝑦)) |
159 | 154 | simprd 496 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦) |
160 | 159 | r19.21bi 3208 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → (𝑐‘𝑦) ≠ 𝑦) |
161 | 158, 160 | eqnetrd 3083 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
162 | 161 | ralrimiva 3182 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
163 | 156 | simp2d 1135 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀) |
164 | | elfzuz 12894 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ∈
(ℤ≥‘2)) |
165 | | eluz2b3 12311 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈
(ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) |
166 | 165 | simprbi 497 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ≠ 1) |
167 | 58, 164, 166 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ≠ 1) |
168 | 167 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ≠ 1) |
169 | 163, 168 | eqnetrd 3083 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠
1) |
170 | 156 | simp3d 1136 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1) |
171 | 168 | necomd 3071 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 1 ≠ 𝑀) |
172 | 170, 171 | eqnetrd 3083 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀) |
173 | | fveq2 6664 |
. . . . . . . . . . . 12
⊢ (𝑦 = 1 → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
174 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑦 = 1 → 𝑦 = 1) |
175 | 173, 174 | neeq12d 3077 |
. . . . . . . . . . 11
⊢ (𝑦 = 1 → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠
1)) |
176 | | fveq2 6664 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
177 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
178 | 176, 177 | neeq12d 3077 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑀 → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀)) |
179 | 71, 59, 175, 178 | ralpr 4630 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈
{1, 𝑀} ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠ 1 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀)) |
180 | 169, 172,
179 | sylanbrc 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
181 | | ralunb 4166 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
(𝐾 ∪ {1, 𝑀})((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ (∀𝑦 ∈ 𝐾 ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ∧ ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
182 | 162, 180,
181 | sylanbrc 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
183 | 61 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
184 | 183 | raleqdv 3416 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
185 | 182, 184 | mpbid 233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
186 | | prex 5324 |
. . . . . . . . 9
⊢ {〈1,
𝑀〉, 〈𝑀, 1〉} ∈
V |
187 | 147, 186 | unex 7457 |
. . . . . . . 8
⊢ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ V |
188 | | f1oeq1 6598 |
. . . . . . . . 9
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) |
189 | | fveq1 6663 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑓‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦)) |
190 | 189 | neeq1d 3075 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
191 | 190 | ralbidv 3197 |
. . . . . . . . 9
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
192 | 188, 191 | anbi12d 630 |
. . . . . . . 8
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦) ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦))) |
193 | 187, 192,
1 | elab2 3669 |
. . . . . . 7
⊢ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐴 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
194 | 157, 185,
193 | sylanbrc 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐴) |
195 | 163, 170 | jca 512 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1)) |
196 | | fveq1 6663 |
. . . . . . . . 9
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑔‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
197 | 196 | eqeq1d 2823 |
. . . . . . . 8
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑔‘1) = 𝑀 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀)) |
198 | | fveq1 6663 |
. . . . . . . . 9
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑔‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
199 | 198 | eqeq1d 2823 |
. . . . . . . 8
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑔‘𝑀) = 1 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1)) |
200 | 197, 199 | anbi12d 630 |
. . . . . . 7
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1))) |
201 | 200, 6 | elrab2 3682 |
. . . . . 6
⊢ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐵 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐴 ∧ (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1))) |
202 | 194, 195,
201 | sylanbrc 583 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐵) |
203 | 202 | ex 413 |
. . . 4
⊢ (𝜑 → (𝑐 ∈ 𝐶 → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐵)) |
204 | 67 | adantrr 713 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = 𝑀) |
205 | 163 | adantrl 712 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀) |
206 | 204, 205 | eqtr4d 2859 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
207 | 70 | adantrr 713 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = 1) |
208 | 170 | adantrl 712 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1) |
209 | 207, 208 | eqtr4d 2859 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
210 | | fveq2 6664 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 1 → (𝑏‘𝑦) = (𝑏‘1)) |
211 | 210, 173 | eqeq12d 2837 |
. . . . . . . . . . . 12
⊢ (𝑦 = 1 → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1))) |
212 | | fveq2 6664 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → (𝑏‘𝑦) = (𝑏‘𝑀)) |
213 | 212, 176 | eqeq12d 2837 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀))) |
214 | 71, 59, 211, 213 | ralpr 4630 |
. . . . . . . . . . 11
⊢
(∀𝑦 ∈
{1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ((𝑏‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ∧ (𝑏‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀))) |
215 | 206, 209,
214 | sylanbrc 583 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦)) |
216 | 215 | biantrud 532 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ∧ ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦)))) |
217 | | ralunb 4166 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
(𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ∧ ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
218 | 216, 217 | syl6bbr 290 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
219 | 158 | eqeq2d 2832 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))) |
220 | 219 | ralbidva 3196 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
221 | 220 | adantrl 712 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
222 | 61 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
223 | 222 | raleqdv 3416 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
224 | 218, 221,
223 | 3bitr3rd 311 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
225 | 135 | eqeq2d 2832 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑐‘𝑦) = (𝑏‘𝑦))) |
226 | | eqcom 2828 |
. . . . . . . . 9
⊢ ((𝑐‘𝑦) = (𝑏‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)) |
227 | 225, 226 | syl6bb 288 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))) |
228 | 227 | ralbiia 3164 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)) |
229 | 224, 228 | syl6bbr 290 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
230 | 47 | adantrr 713 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 Fn (1...(𝑁 + 1))) |
231 | 157 | adantrl 712 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
232 | | f1ofn 6610 |
. . . . . . . 8
⊢ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) Fn (1...(𝑁 + 1))) |
233 | 231, 232 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) Fn (1...(𝑁 + 1))) |
234 | | eqfnfv 6795 |
. . . . . . 7
⊢ ((𝑏 Fn (1...(𝑁 + 1)) ∧ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) Fn (1...(𝑁 + 1))) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
235 | 230, 233,
234 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
236 | 155 | adantrl 712 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐:𝐾–1-1-onto→𝐾) |
237 | | f1ofn 6610 |
. . . . . . . 8
⊢ (𝑐:𝐾–1-1-onto→𝐾 → 𝑐 Fn 𝐾) |
238 | 236, 237 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 Fn 𝐾) |
239 | 127 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐾 ⊆ (1...(𝑁 + 1))) |
240 | | fnssres 6464 |
. . . . . . . 8
⊢ ((𝑏 Fn (1...(𝑁 + 1)) ∧ 𝐾 ⊆ (1...(𝑁 + 1))) → (𝑏 ↾ 𝐾) Fn 𝐾) |
241 | 230, 239,
240 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 ↾ 𝐾) Fn 𝐾) |
242 | | eqfnfv 6795 |
. . . . . . 7
⊢ ((𝑐 Fn 𝐾 ∧ (𝑏 ↾ 𝐾) Fn 𝐾) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
243 | 238, 241,
242 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
244 | 229, 235,
243 | 3bitr4d 312 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ 𝑐 = (𝑏 ↾ 𝐾))) |
245 | 244 | ex 413 |
. . . 4
⊢ (𝜑 → ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ 𝑐 = (𝑏 ↾ 𝐾)))) |
246 | 12, 23, 142, 203, 245 | en3d 8535 |
. . 3
⊢ (𝜑 → 𝐵 ≈ 𝐶) |
247 | | hashen 13697 |
. . . 4
⊢ ((𝐵 ∈ Fin ∧ 𝐶 ∈ Fin) →
((♯‘𝐵) =
(♯‘𝐶) ↔
𝐵 ≈ 𝐶)) |
248 | 10, 21, 247 | mp2an 688 |
. . 3
⊢
((♯‘𝐵) =
(♯‘𝐶) ↔
𝐵 ≈ 𝐶) |
249 | 246, 248 | sylibr 235 |
. 2
⊢ (𝜑 → (♯‘𝐵) = (♯‘𝐶)) |
250 | 13 | fveq2i 6667 |
. . . 4
⊢
(♯‘𝐶) =
(♯‘{𝑓 ∣
(𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}) |
251 | 55 | derangval 32312 |
. . . . 5
⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})) |
252 | 18, 251 | ax-mp 5 |
. . . 4
⊢ (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}) |
253 | 55, 56 | derangen2 32319 |
. . . . 5
⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (𝑆‘(♯‘𝐾))) |
254 | 18, 253 | ax-mp 5 |
. . . 4
⊢ (𝐷‘𝐾) = (𝑆‘(♯‘𝐾)) |
255 | 250, 252,
254 | 3eqtr2ri 2851 |
. . 3
⊢ (𝑆‘(♯‘𝐾)) = (♯‘𝐶) |
256 | 60 | simp3d 1136 |
. . . 4
⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1)) |
257 | 256 | fveq2d 6668 |
. . 3
⊢ (𝜑 → (𝑆‘(♯‘𝐾)) = (𝑆‘(𝑁 − 1))) |
258 | 255, 257 | syl5eqr 2870 |
. 2
⊢ (𝜑 → (♯‘𝐶) = (𝑆‘(𝑁 − 1))) |
259 | 249, 258 | eqtrd 2856 |
1
⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1))) |