Step |
Hyp |
Ref |
Expression |
1 |
|
derang.d |
⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) ) |
2 |
|
subfac.n |
⊢ 𝑆 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐷 ‘ ( 1 ... 𝑛 ) ) ) |
3 |
|
subfacp1lem.a |
⊢ 𝐴 = { 𝑓 ∣ ( 𝑓 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 + 1 ) ) ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } |
4 |
|
subfacp1lem1.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
5 |
|
subfacp1lem1.m |
⊢ ( 𝜑 → 𝑀 ∈ ( 2 ... ( 𝑁 + 1 ) ) ) |
6 |
|
subfacp1lem1.x |
⊢ 𝑀 ∈ V |
7 |
|
subfacp1lem1.k |
⊢ 𝐾 = ( ( 2 ... ( 𝑁 + 1 ) ) ∖ { 𝑀 } ) |
8 |
|
subfacp1lem5.b |
⊢ 𝐵 = { 𝑔 ∈ 𝐴 ∣ ( ( 𝑔 ‘ 1 ) = 𝑀 ∧ ( 𝑔 ‘ 𝑀 ) ≠ 1 ) } |
9 |
|
subfacp1lem5.f |
⊢ 𝐹 = ( ( I ↾ 𝐾 ) ∪ { 〈 1 , 𝑀 〉 , 〈 𝑀 , 1 〉 } ) |
10 |
|
f1oi |
⊢ ( I ↾ 𝐾 ) : 𝐾 –1-1-onto→ 𝐾 |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( I ↾ 𝐾 ) : 𝐾 –1-1-onto→ 𝐾 ) |
12 |
1 2 3 4 5 6 7 9 11
|
subfacp1lem2a |
⊢ ( 𝜑 → ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝐹 ‘ 1 ) = 𝑀 ∧ ( 𝐹 ‘ 𝑀 ) = 1 ) ) |
13 |
12
|
simp1d |
⊢ ( 𝜑 → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) |
14 |
|
f1ocnv |
⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ◡ 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) |
15 |
|
f1ofn |
⊢ ( ◡ 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → ◡ 𝐹 Fn ( 1 ... ( 𝑁 + 1 ) ) ) |
16 |
13 14 15
|
3syl |
⊢ ( 𝜑 → ◡ 𝐹 Fn ( 1 ... ( 𝑁 + 1 ) ) ) |
17 |
|
f1ofn |
⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → 𝐹 Fn ( 1 ... ( 𝑁 + 1 ) ) ) |
18 |
13 17
|
syl |
⊢ ( 𝜑 → 𝐹 Fn ( 1 ... ( 𝑁 + 1 ) ) ) |
19 |
1 2 3 4 5 6 7
|
subfacp1lem1 |
⊢ ( 𝜑 → ( ( 𝐾 ∩ { 1 , 𝑀 } ) = ∅ ∧ ( 𝐾 ∪ { 1 , 𝑀 } ) = ( 1 ... ( 𝑁 + 1 ) ) ∧ ( ♯ ‘ 𝐾 ) = ( 𝑁 − 1 ) ) ) |
20 |
19
|
simp2d |
⊢ ( 𝜑 → ( 𝐾 ∪ { 1 , 𝑀 } ) = ( 1 ... ( 𝑁 + 1 ) ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐾 ∪ { 1 , 𝑀 } ) ↔ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) ) |
22 |
21
|
biimpar |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝑥 ∈ ( 𝐾 ∪ { 1 , 𝑀 } ) ) |
23 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝐾 ∪ { 1 , 𝑀 } ) ↔ ( 𝑥 ∈ 𝐾 ∨ 𝑥 ∈ { 1 , 𝑀 } ) ) |
24 |
22 23
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝑥 ∈ 𝐾 ∨ 𝑥 ∈ { 1 , 𝑀 } ) ) |
25 |
1 2 3 4 5 6 7 9 11
|
subfacp1lem2b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑥 ) = ( ( I ↾ 𝐾 ) ‘ 𝑥 ) ) |
26 |
|
fvresi |
⊢ ( 𝑥 ∈ 𝐾 → ( ( I ↾ 𝐾 ) ‘ 𝑥 ) = 𝑥 ) |
27 |
26
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( ( I ↾ 𝐾 ) ‘ 𝑥 ) = 𝑥 ) |
28 |
25 27
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
29 |
28
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
30 |
29 28
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
31 |
|
vex |
⊢ 𝑥 ∈ V |
32 |
31
|
elpr |
⊢ ( 𝑥 ∈ { 1 , 𝑀 } ↔ ( 𝑥 = 1 ∨ 𝑥 = 𝑀 ) ) |
33 |
12
|
simp2d |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = 𝑀 ) |
34 |
33
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐹 ‘ 1 ) ) = ( 𝐹 ‘ 𝑀 ) ) |
35 |
12
|
simp3d |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = 1 ) |
36 |
34 35
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐹 ‘ 1 ) ) = 1 ) |
37 |
|
2fveq3 |
⊢ ( 𝑥 = 1 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ) |
38 |
|
id |
⊢ ( 𝑥 = 1 → 𝑥 = 1 ) |
39 |
37 38
|
eqeq12d |
⊢ ( 𝑥 = 1 → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ↔ ( 𝐹 ‘ ( 𝐹 ‘ 1 ) ) = 1 ) ) |
40 |
36 39
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑥 = 1 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) ) |
41 |
35
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑀 ) ) = ( 𝐹 ‘ 1 ) ) |
42 |
41 33
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑀 ) ) = 𝑀 ) |
43 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑀 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐹 ‘ 𝑀 ) ) ) |
44 |
|
id |
⊢ ( 𝑥 = 𝑀 → 𝑥 = 𝑀 ) |
45 |
43 44
|
eqeq12d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ↔ ( 𝐹 ‘ ( 𝐹 ‘ 𝑀 ) ) = 𝑀 ) ) |
46 |
42 45
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑥 = 𝑀 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) ) |
47 |
40 46
|
jaod |
⊢ ( 𝜑 → ( ( 𝑥 = 1 ∨ 𝑥 = 𝑀 ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) ) |
48 |
47
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 1 ∨ 𝑥 = 𝑀 ) ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
49 |
32 48
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 1 , 𝑀 } ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
50 |
30 49
|
jaodan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐾 ∨ 𝑥 ∈ { 1 , 𝑀 } ) ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
51 |
24 50
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
52 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ) |
53 |
|
f1of |
⊢ ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ) |
54 |
13 53
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) ⟶ ( 1 ... ( 𝑁 + 1 ) ) ) |
55 |
54
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) ) |
56 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : ( 1 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 1 ... ( 𝑁 + 1 ) ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) ) |
57 |
52 55 56
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) ) |
58 |
51 57
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
59 |
16 18 58
|
eqfnfvd |
⊢ ( 𝜑 → ◡ 𝐹 = 𝐹 ) |