Step |
Hyp |
Ref |
Expression |
1 |
|
yoniso.y |
⊢ 𝑌 = ( Yon ‘ 𝐶 ) |
2 |
|
yoniso.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
3 |
|
yoniso.s |
⊢ 𝑆 = ( SetCat ‘ 𝑈 ) |
4 |
|
yoniso.d |
⊢ 𝐷 = ( CatCat ‘ 𝑉 ) |
5 |
|
yoniso.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
6 |
|
yoniso.i |
⊢ 𝐼 = ( Iso ‘ 𝐷 ) |
7 |
|
yoniso.q |
⊢ 𝑄 = ( 𝑂 FuncCat 𝑆 ) |
8 |
|
yoniso.e |
⊢ 𝐸 = ( 𝑄 ↾s ran ( 1st ‘ 𝑌 ) ) |
9 |
|
yoniso.v |
⊢ ( 𝜑 → 𝑉 ∈ 𝑋 ) |
10 |
|
yoniso.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
11 |
|
yoniso.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑊 ) |
12 |
|
yoniso.h |
⊢ ( 𝜑 → ran ( Homf ‘ 𝐶 ) ⊆ 𝑈 ) |
13 |
|
yoniso.eb |
⊢ ( 𝜑 → 𝐸 ∈ 𝐵 ) |
14 |
|
yoniso.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) = 𝑦 ) |
15 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝑄 ) |
16 |
4 5 9
|
catcbas |
⊢ ( 𝜑 → 𝐵 = ( 𝑉 ∩ Cat ) ) |
17 |
|
inss2 |
⊢ ( 𝑉 ∩ Cat ) ⊆ Cat |
18 |
16 17
|
eqsstrdi |
⊢ ( 𝜑 → 𝐵 ⊆ Cat ) |
19 |
18 10
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
20 |
1 19 2 3 7 11 12
|
yoncl |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐶 Func 𝑄 ) ) |
21 |
|
1st2nd |
⊢ ( ( Rel ( 𝐶 Func 𝑄 ) ∧ 𝑌 ∈ ( 𝐶 Func 𝑄 ) ) → 𝑌 = 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ) |
22 |
15 20 21
|
sylancr |
⊢ ( 𝜑 → 𝑌 = 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ) |
23 |
1 2 3 7 19 11 12
|
yonffth |
⊢ ( 𝜑 → 𝑌 ∈ ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ) |
24 |
22 23
|
eqeltrrd |
⊢ ( 𝜑 → 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
26 |
2
|
oppccat |
⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat ) |
27 |
19 26
|
syl |
⊢ ( 𝜑 → 𝑂 ∈ Cat ) |
28 |
3
|
setccat |
⊢ ( 𝑈 ∈ 𝑊 → 𝑆 ∈ Cat ) |
29 |
11 28
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Cat ) |
30 |
7 27 29
|
fuccat |
⊢ ( 𝜑 → 𝑄 ∈ Cat ) |
31 |
|
fvex |
⊢ ( 1st ‘ 𝑌 ) ∈ V |
32 |
31
|
rnex |
⊢ ran ( 1st ‘ 𝑌 ) ∈ V |
33 |
32
|
a1i |
⊢ ( 𝜑 → ran ( 1st ‘ 𝑌 ) ∈ V ) |
34 |
7
|
fucbas |
⊢ ( 𝑂 Func 𝑆 ) = ( Base ‘ 𝑄 ) |
35 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func 𝑄 ) ∧ 𝑌 ∈ ( 𝐶 Func 𝑄 ) ) → ( 1st ‘ 𝑌 ) ( 𝐶 Func 𝑄 ) ( 2nd ‘ 𝑌 ) ) |
36 |
15 20 35
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) ( 𝐶 Func 𝑄 ) ( 2nd ‘ 𝑌 ) ) |
37 |
25 34 36
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) ⟶ ( 𝑂 Func 𝑆 ) ) |
38 |
37
|
ffnd |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) Fn ( Base ‘ 𝐶 ) ) |
39 |
|
dffn3 |
⊢ ( ( 1st ‘ 𝑌 ) Fn ( Base ‘ 𝐶 ) ↔ ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) ⟶ ran ( 1st ‘ 𝑌 ) ) |
40 |
38 39
|
sylib |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) ⟶ ran ( 1st ‘ 𝑌 ) ) |
41 |
25 8 30 33 40
|
ffthres2c |
⊢ ( 𝜑 → ( ( 1st ‘ 𝑌 ) ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ( 2nd ‘ 𝑌 ) ↔ ( 1st ‘ 𝑌 ) ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ( 2nd ‘ 𝑌 ) ) ) |
42 |
|
df-br |
⊢ ( ( 1st ‘ 𝑌 ) ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ( 2nd ‘ 𝑌 ) ↔ 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ) |
43 |
|
df-br |
⊢ ( ( 1st ‘ 𝑌 ) ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ( 2nd ‘ 𝑌 ) ↔ 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ) |
44 |
41 42 43
|
3bitr3g |
⊢ ( 𝜑 → ( 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝑄 ) ∩ ( 𝐶 Faith 𝑄 ) ) ↔ 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ) ) |
45 |
24 44
|
mpbid |
⊢ ( 𝜑 → 〈 ( 1st ‘ 𝑌 ) , ( 2nd ‘ 𝑌 ) 〉 ∈ ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ) |
46 |
22 45
|
eqeltrd |
⊢ ( 𝜑 → 𝑌 ∈ ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ) |
47 |
|
fveq2 |
⊢ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) = ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ) |
48 |
47
|
fveq1d |
⊢ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) = ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) |
49 |
48
|
fveq2d |
⊢ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) ) |
50 |
|
simpl |
⊢ ( ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
51 |
50 50
|
jca |
⊢ ( ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) |
52 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↔ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) |
53 |
52
|
anbi2d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ↔ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) ) |
54 |
53
|
anbi2d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ↔ ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) ) ) |
55 |
|
2fveq3 |
⊢ ( 𝑦 = 𝑥 → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) = ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
56 |
55
|
fveq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) = ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) |
57 |
56
|
fveq2d |
⊢ ( 𝑦 = 𝑥 → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) ) |
58 |
|
id |
⊢ ( 𝑦 = 𝑥 → 𝑦 = 𝑥 ) |
59 |
57 58
|
eqeq12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) = 𝑦 ↔ ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = 𝑥 ) ) |
60 |
54 59
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) = 𝑦 ) ↔ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = 𝑥 ) ) ) |
61 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → 𝐶 ∈ Cat ) |
62 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) ) |
63 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
64 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
65 |
1 25 61 62 63 64
|
yon11 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) |
66 |
65
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ) |
67 |
66 14
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) = 𝑦 ) |
68 |
60 67
|
chvarvv |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = 𝑥 ) |
69 |
51 68
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = 𝑥 ) |
70 |
69 67
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) ) ‘ 𝑥 ) ) ↔ 𝑥 = 𝑦 ) ) |
71 |
49 70
|
syl5ib |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
72 |
71
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
73 |
|
dff13 |
⊢ ( ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1→ ( 𝑂 Func 𝑆 ) ↔ ( ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) ⟶ ( 𝑂 Func 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝑌 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
74 |
37 72 73
|
sylanbrc |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1→ ( 𝑂 Func 𝑆 ) ) |
75 |
|
f1f1orn |
⊢ ( ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1→ ( 𝑂 Func 𝑆 ) → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ran ( 1st ‘ 𝑌 ) ) |
76 |
74 75
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ran ( 1st ‘ 𝑌 ) ) |
77 |
37
|
frnd |
⊢ ( 𝜑 → ran ( 1st ‘ 𝑌 ) ⊆ ( 𝑂 Func 𝑆 ) ) |
78 |
8 34
|
ressbas2 |
⊢ ( ran ( 1st ‘ 𝑌 ) ⊆ ( 𝑂 Func 𝑆 ) → ran ( 1st ‘ 𝑌 ) = ( Base ‘ 𝐸 ) ) |
79 |
77 78
|
syl |
⊢ ( 𝜑 → ran ( 1st ‘ 𝑌 ) = ( Base ‘ 𝐸 ) ) |
80 |
79
|
f1oeq3d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ran ( 1st ‘ 𝑌 ) ↔ ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐸 ) ) ) |
81 |
76 80
|
mpbid |
⊢ ( 𝜑 → ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐸 ) ) |
82 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
83 |
4 5 25 82 9 10 13 6
|
catciso |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝐶 𝐼 𝐸 ) ↔ ( 𝑌 ∈ ( ( 𝐶 Full 𝐸 ) ∩ ( 𝐶 Faith 𝐸 ) ) ∧ ( 1st ‘ 𝑌 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐸 ) ) ) ) |
84 |
46 81 83
|
mpbir2and |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐶 𝐼 𝐸 ) ) |