Step | Hyp | Ref
| Expression |
1 | | subfacp1lem.a |
. . . . . . 7
⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
2 | | fzfi 13545 |
. . . . . . . 8
⊢
(1...(𝑁 + 1)) ∈
Fin |
3 | | deranglem 32841 |
. . . . . . . 8
⊢
((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 |
. . . . . . 7
⊢ {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin |
5 | 1, 4 | eqeltri 2834 |
. . . . . 6
⊢ 𝐴 ∈ Fin |
6 | | subfacp1lem3.b |
. . . . . . 7
⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)} |
7 | 6 | ssrab3 3995 |
. . . . . 6
⊢ 𝐵 ⊆ 𝐴 |
8 | | ssfi 8851 |
. . . . . 6
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
9 | 5, 7, 8 | mp2an 692 |
. . . . 5
⊢ 𝐵 ∈ Fin |
10 | 9 | elexi 3427 |
. . . 4
⊢ 𝐵 ∈ V |
11 | 10 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐵 ∈ V) |
12 | | eqid 2737 |
. . . 4
⊢ (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)) = (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)) |
13 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵) |
14 | | fveq1 6716 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑏 → (𝑔‘1) = (𝑏‘1)) |
15 | 14 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑏 → ((𝑔‘1) = 𝑀 ↔ (𝑏‘1) = 𝑀)) |
16 | | fveq1 6716 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑏 → (𝑔‘𝑀) = (𝑏‘𝑀)) |
17 | 16 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑏 → ((𝑔‘𝑀) = 1 ↔ (𝑏‘𝑀) = 1)) |
18 | 15, 17 | anbi12d 634 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑏 → (((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1) ↔ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
19 | 18, 6 | elrab2 3605 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
20 | 13, 19 | sylib 221 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ∈ 𝐴 ∧ ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1))) |
21 | 20 | simpld 498 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴) |
22 | | vex 3412 |
. . . . . . . . . . 11
⊢ 𝑏 ∈ V |
23 | | f1oeq1 6649 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑏 → (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ↔ 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))) |
24 | | fveq1 6716 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑏 → (𝑓‘𝑦) = (𝑏‘𝑦)) |
25 | 24 | neeq1d 3000 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑏 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦)) |
26 | 25 | ralbidv 3118 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑏 → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
27 | 23, 26 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (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))) |
28 | 22, 27, 1 | elab2 3591 |
. . . . . . . . . 10
⊢ (𝑏 ∈ 𝐴 ↔ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
29 | 21, 28 | sylib 221 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦)) |
30 | 29 | simpld 498 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
31 | | f1of1 6660 |
. . . . . . . 8
⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1))) |
32 | | df-f1 6385 |
. . . . . . . . 9
⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) ↔ (𝑏:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)) ∧ Fun ◡𝑏)) |
33 | 32 | simprbi 500 |
. . . . . . . 8
⊢ (𝑏:(1...(𝑁 + 1))–1-1→(1...(𝑁 + 1)) → Fun ◡𝑏) |
34 | 30, 31, 33 | 3syl 18 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Fun ◡𝑏) |
35 | | f1ofn 6662 |
. . . . . . . . . . 11
⊢ (𝑏:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝑏 Fn (1...(𝑁 + 1))) |
36 | 30, 35 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 Fn (1...(𝑁 + 1))) |
37 | | fnresdm 6496 |
. . . . . . . . . 10
⊢ (𝑏 Fn (1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))) = 𝑏) |
38 | | f1oeq1 6649 |
. . . . . . . . . 10
⊢ ((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)))) |
39 | 36, 37, 38 | 3syl 18 |
. . . . . . . . 9
⊢ ((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)))) |
40 | 30, 39 | mpbird 260 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
41 | | f1ofo 6668 |
. . . . . . . 8
⊢ ((𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1))) |
42 | 40, 41 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ (1...(𝑁 + 1))):(1...(𝑁 + 1))–onto→(1...(𝑁 + 1))) |
43 | | ssun2 4087 |
. . . . . . . . . . . 12
⊢ {1, 𝑀} ⊆ (𝐾 ∪ {1, 𝑀}) |
44 | | derang.d |
. . . . . . . . . . . . . 14
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
45 | | subfac.n |
. . . . . . . . . . . . . 14
⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
46 | | subfacp1lem1.n |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℕ) |
47 | | subfacp1lem1.m |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) |
48 | | subfacp1lem1.x |
. . . . . . . . . . . . . 14
⊢ 𝑀 ∈ V |
49 | | subfacp1lem1.k |
. . . . . . . . . . . . . 14
⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) |
50 | 44, 45, 1, 46, 47, 48, 49 | subfacp1lem1 32854 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1))) |
51 | 50 | simp2d 1145 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
52 | 43, 51 | sseqtrid 3953 |
. . . . . . . . . . 11
⊢ (𝜑 → {1, 𝑀} ⊆ (1...(𝑁 + 1))) |
53 | 52 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → {1, 𝑀} ⊆ (1...(𝑁 + 1))) |
54 | 36, 53 | fnssresd 6501 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀}) |
55 | 20 | simprd 499 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑏‘1) = 𝑀 ∧ (𝑏‘𝑀) = 1)) |
56 | 55 | simpld 498 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) = 𝑀) |
57 | 48 | prid2 4679 |
. . . . . . . . . . . 12
⊢ 𝑀 ∈ {1, 𝑀} |
58 | 56, 57 | eqeltrdi 2846 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘1) ∈ {1, 𝑀}) |
59 | 55 | simprd 499 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) = 1) |
60 | | 1ex 10829 |
. . . . . . . . . . . . 13
⊢ 1 ∈
V |
61 | 60 | prid1 4678 |
. . . . . . . . . . . 12
⊢ 1 ∈
{1, 𝑀} |
62 | 59, 61 | eqeltrdi 2846 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏‘𝑀) ∈ {1, 𝑀}) |
63 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 1 → (𝑏‘𝑥) = (𝑏‘1)) |
64 | 63 | eleq1d 2822 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘1) ∈ {1, 𝑀})) |
65 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑀 → (𝑏‘𝑥) = (𝑏‘𝑀)) |
66 | 65 | eleq1d 2822 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑀 → ((𝑏‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑀) ∈ {1, 𝑀})) |
67 | 60, 48, 64, 66 | ralpr 4616 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
{1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀} ↔ ((𝑏‘1) ∈ {1, 𝑀} ∧ (𝑏‘𝑀) ∈ {1, 𝑀})) |
68 | 58, 62, 67 | sylanbrc 586 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀}) |
69 | | fvres 6736 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑥) = (𝑏‘𝑥)) |
70 | 69 | eleq1d 2822 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ (𝑏‘𝑥) ∈ {1, 𝑀})) |
71 | 70 | ralbiia 3087 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
{1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀} ↔ ∀𝑥 ∈ {1, 𝑀} (𝑏‘𝑥) ∈ {1, 𝑀}) |
72 | 68, 71 | sylibr 237 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀}) |
73 | | ffnfv 6935 |
. . . . . . . . 9
⊢ ((𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀} ↔ ((𝑏 ↾ {1, 𝑀}) Fn {1, 𝑀} ∧ ∀𝑥 ∈ {1, 𝑀} ((𝑏 ↾ {1, 𝑀})‘𝑥) ∈ {1, 𝑀})) |
74 | 54, 72, 73 | sylanbrc 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}⟶{1, 𝑀}) |
75 | | fveqeq2 6726 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = 1 ↔ (𝑏‘𝑀) = 1)) |
76 | 75 | rspcev 3537 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ {1, 𝑀} ∧ (𝑏‘𝑀) = 1) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1) |
77 | 57, 59, 76 | sylancr 590 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1) |
78 | | fveqeq2 6726 |
. . . . . . . . . . . 12
⊢ (𝑦 = 1 → ((𝑏‘𝑦) = 𝑀 ↔ (𝑏‘1) = 𝑀)) |
79 | 78 | rspcev 3537 |
. . . . . . . . . . 11
⊢ ((1
∈ {1, 𝑀} ∧ (𝑏‘1) = 𝑀) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀) |
80 | 61, 56, 79 | sylancr 590 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀) |
81 | | eqeq2 2749 |
. . . . . . . . . . . 12
⊢ (𝑥 = 1 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 1)) |
82 | 81 | rexbidv 3216 |
. . . . . . . . . . 11
⊢ (𝑥 = 1 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1)) |
83 | | eqeq2 2749 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑀 → ((𝑏‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑀)) |
84 | 83 | rexbidv 3216 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)) |
85 | 60, 48, 82, 84 | ralpr 4616 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
{1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥 ↔ (∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 1 ∧ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑀)) |
86 | 77, 80, 85 | sylanbrc 586 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
87 | | eqcom 2744 |
. . . . . . . . . . . 12
⊢ (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥) |
88 | | fvres 6736 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {1, 𝑀} → ((𝑏 ↾ {1, 𝑀})‘𝑦) = (𝑏‘𝑦)) |
89 | 88 | eqeq1d 2739 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {1, 𝑀} → (((𝑏 ↾ {1, 𝑀})‘𝑦) = 𝑥 ↔ (𝑏‘𝑦) = 𝑥)) |
90 | 87, 89 | syl5bb 286 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ {1, 𝑀} → (𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ (𝑏‘𝑦) = 𝑥)) |
91 | 90 | rexbiia 3169 |
. . . . . . . . . 10
⊢
(∃𝑦 ∈ {1,
𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
92 | 91 | ralbii 3088 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
{1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦) ↔ ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = 𝑥) |
93 | 86, 92 | sylibr 237 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑥 ∈ {1, 𝑀}∃𝑦 ∈ {1, 𝑀}𝑥 = ((𝑏 ↾ {1, 𝑀})‘𝑦)) |
94 | | dffo3 6921 |
. . . . . . . 8
⊢ ((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))) |
95 | 74, 93, 94 | sylanbrc 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ {1, 𝑀}):{1, 𝑀}–onto→{1, 𝑀}) |
96 | | resdif 6681 |
. . . . . . 7
⊢ ((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})) |
97 | 34, 42, 95, 96 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀})) |
98 | | uncom 4067 |
. . . . . . . . . 10
⊢ ({1,
𝑀} ∪ 𝐾) = (𝐾 ∪ {1, 𝑀}) |
99 | 98, 51 | syl5eq 2790 |
. . . . . . . . 9
⊢ (𝜑 → ({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1))) |
100 | | incom 4115 |
. . . . . . . . . . 11
⊢ ({1,
𝑀} ∩ 𝐾) = (𝐾 ∩ {1, 𝑀}) |
101 | 50 | simp1d 1144 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅) |
102 | 100, 101 | syl5eq 2790 |
. . . . . . . . . 10
⊢ (𝜑 → ({1, 𝑀} ∩ 𝐾) = ∅) |
103 | | uneqdifeq 4404 |
. . . . . . . . . 10
⊢ (({1,
𝑀} ⊆ (1...(𝑁 + 1)) ∧ ({1, 𝑀} ∩ 𝐾) = ∅) → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)) |
104 | 52, 102, 103 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (({1, 𝑀} ∪ 𝐾) = (1...(𝑁 + 1)) ↔ ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾)) |
105 | 99, 104 | mpbid 235 |
. . . . . . . 8
⊢ (𝜑 → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾) |
106 | 105 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((1...(𝑁 + 1)) ∖ {1, 𝑀}) = 𝐾) |
107 | | reseq2 5846 |
. . . . . . . . 9
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → (𝑏 ↾ ((1...(𝑁 + 1)) ∖ {1, 𝑀})) = (𝑏 ↾ 𝐾)) |
108 | 107 | f1oeq1d 6656 |
. . . . . . . 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):((1...(𝑁 + 1)) ∖ {1, 𝑀})–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}))) |
109 | | f1oeq2 6650 |
. . . . . . . 8
⊢
(((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}))) |
110 | | f1oeq3 6651 |
. . . . . . . 8
⊢
(((1...(𝑁 + 1))
∖ {1, 𝑀}) = 𝐾 → ((𝑏 ↾ 𝐾):𝐾–1-1-onto→((1...(𝑁 + 1)) ∖ {1, 𝑀}) ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
111 | 108, 109,
110 | 3bitrd 308 |
. . . . . . 7
⊢
(((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)) |
112 | 106, 111 | syl 17 |
. . . . . 6
⊢ ((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)) |
113 | 97, 112 | mpbid 235 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾) |
114 | | ssun1 4086 |
. . . . . . . 8
⊢ 𝐾 ⊆ (𝐾 ∪ {1, 𝑀}) |
115 | 114, 51 | sseqtrid 3953 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ⊆ (1...(𝑁 + 1))) |
116 | 115 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐾 ⊆ (1...(𝑁 + 1))) |
117 | 29 | simprd 499 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦) |
118 | | ssralv 3967 |
. . . . . 6
⊢ (𝐾 ⊆ (1...(𝑁 + 1)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) ≠ 𝑦 → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
119 | 116, 117,
118 | sylc 65 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦) |
120 | 22 | resex 5899 |
. . . . . 6
⊢ (𝑏 ↾ 𝐾) ∈ V |
121 | | f1oeq1 6649 |
. . . . . . 7
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓:𝐾–1-1-onto→𝐾 ↔ (𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾)) |
122 | | fveq1 6716 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (𝑓‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦)) |
123 | | fvres 6736 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐾 → ((𝑏 ↾ 𝐾)‘𝑦) = (𝑏‘𝑦)) |
124 | 122, 123 | sylan9eq 2798 |
. . . . . . . . 9
⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → (𝑓‘𝑦) = (𝑏‘𝑦)) |
125 | 124 | neeq1d 3000 |
. . . . . . . 8
⊢ ((𝑓 = (𝑏 ↾ 𝐾) ∧ 𝑦 ∈ 𝐾) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑏‘𝑦) ≠ 𝑦)) |
126 | 125 | ralbidva 3117 |
. . . . . . 7
⊢ (𝑓 = (𝑏 ↾ 𝐾) → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
127 | 121, 126 | anbi12d 634 |
. . . . . 6
⊢ (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))) |
128 | | subfacp1lem3.c |
. . . . . 6
⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} |
129 | 120, 127,
128 | elab2 3591 |
. . . . 5
⊢ ((𝑏 ↾ 𝐾) ∈ 𝐶 ↔ ((𝑏 ↾ 𝐾):𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) ≠ 𝑦)) |
130 | 113, 119,
129 | sylanbrc 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏 ↾ 𝐾) ∈ 𝐶) |
131 | 46 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑁 ∈ ℕ) |
132 | 47 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ∈ (2...(𝑁 + 1))) |
133 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) |
134 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶) |
135 | | vex 3412 |
. . . . . . . . . . 11
⊢ 𝑐 ∈ V |
136 | | f1oeq1 6649 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑐 → (𝑓:𝐾–1-1-onto→𝐾 ↔ 𝑐:𝐾–1-1-onto→𝐾)) |
137 | | fveq1 6716 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑐 → (𝑓‘𝑦) = (𝑐‘𝑦)) |
138 | 137 | neeq1d 3000 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑐 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑐‘𝑦) ≠ 𝑦)) |
139 | 138 | ralbidv 3118 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑐 → (∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
140 | 136, 139 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑐 → ((𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦) ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦))) |
141 | 135, 140,
128 | elab2 3591 |
. . . . . . . . . 10
⊢ (𝑐 ∈ 𝐶 ↔ (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
142 | 134, 141 | sylib 221 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦)) |
143 | 142 | simpld 498 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐:𝐾–1-1-onto→𝐾) |
144 | 44, 45, 1, 131, 132, 48, 49, 133, 143 | subfacp1lem2a 32855 |
. . . . . . 7
⊢ ((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)) |
145 | 144 | simp1d 1144 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
146 | 44, 45, 1, 131, 132, 48, 49, 133, 143 | subfacp1lem2b 32856 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = (𝑐‘𝑦)) |
147 | 142 | simprd 499 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) ≠ 𝑦) |
148 | 147 | r19.21bi 3130 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → (𝑐‘𝑦) ≠ 𝑦) |
149 | 146, 148 | eqnetrd 3008 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
150 | 149 | ralrimiva 3105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ 𝐾 ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
151 | 144 | simp2d 1145 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀) |
152 | | elfzuz 13108 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ∈
(ℤ≥‘2)) |
153 | | eluz2b3 12518 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈
(ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1)) |
154 | 153 | simprbi 500 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ≠ 1) |
155 | 47, 152, 154 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ≠ 1) |
156 | 155 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑀 ≠ 1) |
157 | 151, 156 | eqnetrd 3008 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠
1) |
158 | 144 | simp3d 1146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1) |
159 | 156 | necomd 2996 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 1 ≠ 𝑀) |
160 | 158, 159 | eqnetrd 3008 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀) |
161 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑦 = 1 → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
162 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑦 = 1 → 𝑦 = 1) |
163 | 161, 162 | neeq12d 3002 |
. . . . . . . . . 10
⊢ (𝑦 = 1 → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠
1)) |
164 | | fveq2 6717 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑀 → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
165 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
166 | 164, 165 | neeq12d 3002 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑀 → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀)) |
167 | 60, 48, 163, 166 | ralpr 4616 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
{1, 𝑀} ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦 ↔ (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) ≠ 1 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) ≠ 𝑀)) |
168 | 157, 160,
167 | sylanbrc 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ {1, 𝑀} ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
169 | | ralunb 4105 |
. . . . . . . 8
⊢
(∀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)) |
170 | 150, 168,
169 | sylanbrc 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (𝐾 ∪ {1, 𝑀})((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
171 | 51 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
172 | 171 | raleqdv 3325 |
. . . . . . 7
⊢ ((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)) |
173 | 170, 172 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦) |
174 | | prex 5325 |
. . . . . . . 8
⊢ {〈1,
𝑀〉, 〈𝑀, 1〉} ∈
V |
175 | 135, 174 | unex 7531 |
. . . . . . 7
⊢ (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ V |
176 | | f1oeq1 6649 |
. . . . . . . 8
⊢ (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)))) |
177 | | fveq1 6716 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑓‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦)) |
178 | 177 | neeq1d 3000 |
. . . . . . . . 9
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑓‘𝑦) ≠ 𝑦 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
179 | 178 | ralbidv 3118 |
. . . . . . . 8
⊢ (𝑓 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ (1...(𝑁 + 1))((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ≠ 𝑦)) |
180 | 176, 179 | anbi12d 634 |
. . . . . . 7
⊢ (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))) |
181 | 175, 180,
1 | elab2 3591 |
. . . . . 6
⊢ ((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)) |
182 | 145, 173,
181 | sylanbrc 586 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐴) |
183 | 151, 158 | jca 515 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀 ∧ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1)) |
184 | | fveq1 6716 |
. . . . . . . 8
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑔‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
185 | 184 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑔‘1) = 𝑀 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀)) |
186 | | fveq1 6716 |
. . . . . . . 8
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → (𝑔‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
187 | 186 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑔 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) → ((𝑔‘𝑀) = 1 ↔ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1)) |
188 | 185, 187 | anbi12d 634 |
. . . . . 6
⊢ (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))) |
189 | 188, 6 | elrab2 3605 |
. . . . 5
⊢ ((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))) |
190 | 182, 183,
189 | sylanbrc 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ∈ 𝐵) |
191 | 56 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = 𝑀) |
192 | 151 | adantrl 716 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1) = 𝑀) |
193 | 191, 192 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1)) |
194 | 59 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = 1) |
195 | 158 | adantrl 716 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀) = 1) |
196 | 194, 195 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀)) |
197 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ (𝑦 = 1 → (𝑏‘𝑦) = (𝑏‘1)) |
198 | 197, 161 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑦 = 1 → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘1) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘1))) |
199 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → (𝑏‘𝑦) = (𝑏‘𝑀)) |
200 | 199, 164 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑀 → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘𝑀) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑀))) |
201 | 60, 48, 198, 200 | ralpr 4616 |
. . . . . . . . . 10
⊢
(∀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))) |
202 | 193, 196,
201 | sylanbrc 586 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ∀𝑦 ∈ {1, 𝑀} (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦)) |
203 | 202 | biantrud 535 |
. . . . . . . 8
⊢ ((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)))) |
204 | | ralunb 4105 |
. . . . . . . 8
⊢
(∀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))) |
205 | 203, 204 | bitr4di 292 |
. . . . . . 7
⊢ ((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))) |
206 | 146 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐 ∈ 𝐶) ∧ 𝑦 ∈ 𝐾) → ((𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))) |
207 | 206 | ralbidva 3117 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
208 | 207 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
209 | 51 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1))) |
210 | 209 | raleqdv 3325 |
. . . . . . 7
⊢ ((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))) |
211 | 205, 208,
210 | 3bitr3rd 313 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦))) |
212 | 123 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑐‘𝑦) = (𝑏‘𝑦))) |
213 | | eqcom 2744 |
. . . . . . . 8
⊢ ((𝑐‘𝑦) = (𝑏‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦)) |
214 | 212, 213 | bitrdi 290 |
. . . . . . 7
⊢ (𝑦 ∈ 𝐾 → ((𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ (𝑏‘𝑦) = (𝑐‘𝑦))) |
215 | 214 | ralbiia 3087 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑏‘𝑦) = (𝑐‘𝑦)) |
216 | 211, 215 | bitr4di 292 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
217 | 36 | adantrr 717 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 Fn (1...(𝑁 + 1))) |
218 | 145 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
219 | | f1ofn 6662 |
. . . . . . 7
⊢ ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}):(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) Fn (1...(𝑁 + 1))) |
220 | 218, 219 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) Fn (1...(𝑁 + 1))) |
221 | | eqfnfv 6852 |
. . . . . 6
⊢ ((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))) |
222 | 217, 220,
221 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑏‘𝑦) = ((𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉})‘𝑦))) |
223 | 143 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐:𝐾–1-1-onto→𝐾) |
224 | | f1ofn 6662 |
. . . . . . 7
⊢ (𝑐:𝐾–1-1-onto→𝐾 → 𝑐 Fn 𝐾) |
225 | 223, 224 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 Fn 𝐾) |
226 | 115 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝐾 ⊆ (1...(𝑁 + 1))) |
227 | 217, 226 | fnssresd 6501 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 ↾ 𝐾) Fn 𝐾) |
228 | | eqfnfv 6852 |
. . . . . 6
⊢ ((𝑐 Fn 𝐾 ∧ (𝑏 ↾ 𝐾) Fn 𝐾) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
229 | 225, 227,
228 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑐 = (𝑏 ↾ 𝐾) ↔ ∀𝑦 ∈ 𝐾 (𝑐‘𝑦) = ((𝑏 ↾ 𝐾)‘𝑦))) |
230 | 216, 222,
229 | 3bitr4d 314 |
. . . 4
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑏 = (𝑐 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ↔ 𝑐 = (𝑏 ↾ 𝐾))) |
231 | 12, 130, 190, 230 | f1o2d 7459 |
. . 3
⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ (𝑏 ↾ 𝐾)):𝐵–1-1-onto→𝐶) |
232 | 11, 231 | hasheqf1od 13920 |
. 2
⊢ (𝜑 → (♯‘𝐵) = (♯‘𝐶)) |
233 | 128 | fveq2i 6720 |
. . . 4
⊢
(♯‘𝐶) =
(♯‘{𝑓 ∣
(𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}) |
234 | | fzfi 13545 |
. . . . . . 7
⊢
(2...(𝑁 + 1)) ∈
Fin |
235 | | diffi 8906 |
. . . . . . 7
⊢
((2...(𝑁 + 1))
∈ Fin → ((2...(𝑁
+ 1)) ∖ {𝑀}) ∈
Fin) |
236 | 234, 235 | ax-mp 5 |
. . . . . 6
⊢
((2...(𝑁 + 1))
∖ {𝑀}) ∈
Fin |
237 | 49, 236 | eqeltri 2834 |
. . . . 5
⊢ 𝐾 ∈ Fin |
238 | 44 | derangval 32842 |
. . . . 5
⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)})) |
239 | 237, 238 | ax-mp 5 |
. . . 4
⊢ (𝐷‘𝐾) = (♯‘{𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)}) |
240 | 44, 45 | derangen2 32849 |
. . . . 5
⊢ (𝐾 ∈ Fin → (𝐷‘𝐾) = (𝑆‘(♯‘𝐾))) |
241 | 237, 240 | ax-mp 5 |
. . . 4
⊢ (𝐷‘𝐾) = (𝑆‘(♯‘𝐾)) |
242 | 233, 239,
241 | 3eqtr2ri 2772 |
. . 3
⊢ (𝑆‘(♯‘𝐾)) = (♯‘𝐶) |
243 | 50 | simp3d 1146 |
. . . 4
⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1)) |
244 | 243 | fveq2d 6721 |
. . 3
⊢ (𝜑 → (𝑆‘(♯‘𝐾)) = (𝑆‘(𝑁 − 1))) |
245 | 242, 244 | eqtr3id 2792 |
. 2
⊢ (𝜑 → (♯‘𝐶) = (𝑆‘(𝑁 − 1))) |
246 | 232, 245 | eqtrd 2777 |
1
⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1))) |