Step | Hyp | Ref
| Expression |
1 | | relfunc 16522 |
. . 3
⊢ Rel
(𝐶 Func 𝑄) |
2 | | yoneda.y |
. . . 4
⊢ 𝑌 = (Yon‘𝐶) |
3 | | yoneda.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Cat) |
4 | | yoneda.o |
. . . 4
⊢ 𝑂 = (oppCat‘𝐶) |
5 | | yoneda.s |
. . . 4
⊢ 𝑆 = (SetCat‘𝑈) |
6 | | yoneda.q |
. . . 4
⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
7 | | yoneda.w |
. . . . 5
⊢ (𝜑 → 𝑉 ∈ 𝑊) |
8 | | yoneda.v |
. . . . . 6
⊢ (𝜑 → (ran
(Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
9 | 8 | unssbd 3791 |
. . . . 5
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
10 | 7, 9 | ssexd 4805 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ V) |
11 | | yoneda.u |
. . . 4
⊢ (𝜑 → ran
(Homf ‘𝐶) ⊆ 𝑈) |
12 | 2, 3, 4, 5, 6, 10,
11 | yoncl 16902 |
. . 3
⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
13 | | 1st2nd 7214 |
. . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = 〈(1st ‘𝑌), (2nd ‘𝑌)〉) |
14 | 1, 12, 13 | sylancr 695 |
. 2
⊢ (𝜑 → 𝑌 = 〈(1st ‘𝑌), (2nd ‘𝑌)〉) |
15 | | 1st2ndbr 7217 |
. . . . 5
⊢ ((Rel
(𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
16 | 1, 12, 15 | sylancr 695 |
. . . 4
⊢ (𝜑 → (1st
‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
17 | | yoneda.b |
. . . . . . . . . . . . 13
⊢ 𝐵 = (Base‘𝐶) |
18 | 6 | fucbas 16620 |
. . . . . . . . . . . . 13
⊢ (𝑂 Func 𝑆) = (Base‘𝑄) |
19 | 17, 18, 16 | funcf1 16526 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
20 | 19 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
21 | | simprr 796 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵) |
22 | 20, 21 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) |
23 | | simprl 794 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵) |
24 | | opelxpi 5148 |
. . . . . . . . . 10
⊢
((((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆) ∧ 𝑧 ∈ 𝐵) → 〈((1st ‘𝑌)‘𝑤), 𝑧〉 ∈ ((𝑂 Func 𝑆) × 𝐵)) |
25 | 22, 23, 24 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 〈((1st
‘𝑌)‘𝑤), 𝑧〉 ∈ ((𝑂 Func 𝑆) × 𝐵)) |
26 | | yoneda.r |
. . . . . . . . . . . . . 14
⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
27 | 26 | fucbas 16620 |
. . . . . . . . . . . . 13
⊢ ((𝑄 ×c
𝑂) Func 𝑇) = (Base‘𝑅) |
28 | | yonedainv.i |
. . . . . . . . . . . . 13
⊢ 𝐼 = (Inv‘𝑅) |
29 | | yoneda.1 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 =
(Id‘𝐶) |
30 | | yoneda.t |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑇 = (SetCat‘𝑉) |
31 | | yoneda.h |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐻 =
(HomF‘𝑄) |
32 | | yoneda.e |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐸 = (𝑂 evalF 𝑆) |
33 | | yoneda.z |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑍 = (𝐻 ∘func
((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉
∘func (𝑄 2ndF 𝑂))
〈,〉F (𝑄 1stF 𝑂))) |
34 | 2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8 | yonedalem1 16912 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
35 | 34 | simpld 475 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
36 | | funcrcl 16523 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) |
37 | 35, 36 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) |
38 | 37 | simpld 475 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
39 | 37 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ∈ Cat) |
40 | 26, 38, 39 | fuccat 16630 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈ Cat) |
41 | 34 | simprd 479 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
42 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(Iso‘𝑅) =
(Iso‘𝑅) |
43 | | yoneda.m |
. . . . . . . . . . . . . 14
⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) |
44 | | yonedainv.n |
. . . . . . . . . . . . . 14
⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) |
45 | 2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8,
43, 28, 44 | yonedainv 16921 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁) |
46 | 27, 28, 40, 35, 41, 42, 45 | inviso2 16427 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍)) |
47 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (𝑄 ×c
𝑂) = (𝑄 ×c 𝑂) |
48 | 4, 17 | oppcbas 16378 |
. . . . . . . . . . . . . 14
⊢ 𝐵 = (Base‘𝑂) |
49 | 47, 18, 48 | xpcbas 16818 |
. . . . . . . . . . . . 13
⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂)) |
50 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ ((𝑄 ×c
𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇) |
51 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(Iso‘𝑇) =
(Iso‘𝑇) |
52 | 26, 49, 50, 41, 35, 42, 51 | fuciso 16635 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))))) |
53 | 46, 52 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))) |
54 | 53 | simprd 479 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))) |
55 | 54 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))) |
56 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (𝑁‘𝑣) = (𝑁‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
57 | | df-ov 6653 |
. . . . . . . . . . . 12
⊢
(((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
58 | 56, 57 | syl6eqr 2674 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧)) |
59 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
60 | | df-ov 6653 |
. . . . . . . . . . . . 13
⊢
(((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
61 | 59, 60 | syl6eqr 2674 |
. . . . . . . . . . . 12
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)) |
62 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉)) |
63 | | df-ov 6653 |
. . . . . . . . . . . . 13
⊢
(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘〈((1st ‘𝑌)‘𝑤), 𝑧〉) |
64 | 62, 63 | syl6eqr 2674 |
. . . . . . . . . . . 12
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)) |
65 | 61, 64 | oveq12d 6668 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) = ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))) |
66 | 58, 65 | eleq12d 2695 |
. . . . . . . . . 10
⊢ (𝑣 = 〈((1st
‘𝑌)‘𝑤), 𝑧〉 → ((𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))) |
67 | 66 | rspcv 3305 |
. . . . . . . . 9
⊢
(〈((1st ‘𝑌)‘𝑤), 𝑧〉 ∈ ((𝑂 Func 𝑆) × 𝐵) → (∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))) |
68 | 25, 55, 67 | sylc 65 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))) |
69 | 4 | oppccat 16382 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
70 | 3, 69 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑂 ∈ Cat) |
71 | 70 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat) |
72 | 5 | setccat 16735 |
. . . . . . . . . . . . 13
⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
73 | 10, 72 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ Cat) |
74 | 73 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat) |
75 | 32, 71, 74, 48, 22, 23 | evlf1 16860 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘((1st
‘𝑌)‘𝑤))‘𝑧)) |
76 | 3 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat) |
77 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
78 | 2, 17, 76, 21, 77, 23 | yon11 16904 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤)) |
79 | 75, 78 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤)) |
80 | 7 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊) |
81 | 11 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf
‘𝐶) ⊆ 𝑈) |
82 | 8 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
83 | 2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 76, 80, 81, 82, 22, 23 | yonedalem21 16913 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
84 | 79, 83 | oveq12d 6668 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
85 | 68, 84 | eleqtrd 2703 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
86 | 9 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉) |
87 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(Base‘𝑆) =
(Base‘𝑆) |
88 | | relfunc 16522 |
. . . . . . . . . . . . . 14
⊢ Rel
(𝑂 Func 𝑆) |
89 | | 1st2ndbr 7217 |
. . . . . . . . . . . . . 14
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) |
90 | 88, 22, 89 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st
‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑤))) |
91 | 48, 87, 90 | funcf1 16526 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)) |
92 | 91, 23 | ffvelrnd 6360 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆)) |
93 | 5, 10 | setcbas 16728 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
94 | 93 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆)) |
95 | 92, 94 | eleqtrrd 2704 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈) |
96 | 78, 95 | eqeltrrd 2702 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈) |
97 | 86, 96 | sseldd 3604 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉) |
98 | | eqid 2622 |
. . . . . . . . . 10
⊢
(Homf ‘𝑄) = (Homf ‘𝑄) |
99 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆) |
100 | 6, 99 | fuchom 16621 |
. . . . . . . . . 10
⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄) |
101 | 20, 23 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) |
102 | 98, 18, 100, 101, 22 | homfval 16352 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
103 | 8 | unssad 3790 |
. . . . . . . . . . 11
⊢ (𝜑 → ran
(Homf ‘𝑄) ⊆ 𝑉) |
104 | 103 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf
‘𝑄) ⊆ 𝑉) |
105 | 98, 18 | homffn 16353 |
. . . . . . . . . . . 12
⊢
(Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) |
106 | 105 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (Homf
‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))) |
107 | | fnovrn 6809 |
. . . . . . . . . . 11
⊢
(((Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf
‘𝑄)) |
108 | 106, 101,
22, 107 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf
‘𝑄)) |
109 | 104, 108 | sseldd 3604 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉) |
110 | 102, 109 | eqeltrrd 2702 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉) |
111 | 30, 80, 97, 110, 51 | setciso 16741 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
112 | 85, 111 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
113 | 76 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat) |
114 | 113 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat) |
115 | 23 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵) |
116 | 115 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵) |
117 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) |
118 | 2, 17, 114, 116, 77, 117 | yon11 16904 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧)) |
119 | 118 | eqcomd 2628 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)) |
120 | 114 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat) |
121 | 21 | ad3antrrr 766 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵) |
122 | 116 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵) |
123 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐶) =
(comp‘𝐶) |
124 | 117 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵) |
125 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) |
126 | | simpllr 799 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) |
127 | 2, 17, 120, 121, 77, 122, 123, 124, 125, 126 | yon12 16905 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = (ℎ(〈𝑦, 𝑧〉(comp‘𝐶)𝑤)𝑔)) |
128 | 2, 17, 120, 122, 77, 121, 123, 124, 126, 125 | yon2 16906 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔) = (ℎ(〈𝑦, 𝑧〉(comp‘𝐶)𝑤)𝑔)) |
129 | 127, 128 | eqtr4d 2659 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)) |
130 | 119, 129 | mpteq12dva 4732 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (𝑔 ∈ ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔))) |
131 | 16 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
132 | 17, 77, 100, 131, 23, 21 | funcf2 16528 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
133 | 132 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
134 | 99, 133 | nat1st2nd 16611 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st
‘𝑌)‘𝑧))〉(𝑂 Nat 𝑆)〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉)) |
135 | 134 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (〈(1st
‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st
‘𝑌)‘𝑧))〉(𝑂 Nat 𝑆)〈(1st
‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st
‘𝑌)‘𝑤))〉)) |
136 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) |
137 | 99, 135, 48, 136, 117 | natcl 16613 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑦))) |
138 | 10 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V) |
139 | 138 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V) |
140 | 19 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆)) |
141 | 140, 115 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) |
142 | | 1st2ndbr 7217 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑧))) |
143 | 88, 141, 142 | sylancr 695 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑧))) |
144 | 48, 87, 143 | funcf1 16526 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆)) |
145 | 144 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆)) |
146 | 94 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆)) |
147 | 145, 146 | eleqtrrd 2704 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈) |
148 | 91 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st
‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆)) |
149 | 148 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆)) |
150 | 149, 146 | eleqtrrd 2704 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ 𝑈) |
151 | 5, 139, 136, 147, 150 | elsetchom 16731 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦))) |
152 | 137, 151 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st
‘𝑌)‘𝑧))‘𝑦)⟶((1st
‘((1st ‘𝑌)‘𝑤))‘𝑦)) |
153 | 152 | feqmptd 6249 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) = (𝑔 ∈ ((1st
‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔))) |
154 | 130, 153 | eqtr4d 2659 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)) |
155 | 154 | mpteq2dva 4744 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
156 | 80 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊) |
157 | 81 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf
‘𝐶) ⊆ 𝑈) |
158 | 82 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
159 | 22 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) |
160 | 78 | eleq2d 2687 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))) |
161 | 160 | biimpar 502 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ ((1st
‘((1st ‘𝑌)‘𝑤))‘𝑧)) |
162 | 2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 113, 156, 157, 158, 159, 115, 44, 161 | yonedalem4a 16915 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st
‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)))) |
163 | 99, 134, 48 | natfn 16614 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵) |
164 | | dffn5 6241 |
. . . . . . . . . . 11
⊢ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵 ↔ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
165 | 163, 164 | sylib 208 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))) |
166 | 155, 162,
165 | 3eqtr4d 2666 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ)) |
167 | 166 | mpteq2dva 4744 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))) |
168 | | f1of 6137 |
. . . . . . . . . 10
⊢
((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
169 | 112, 168 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
170 | 169 | feqmptd 6249 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ))) |
171 | 132 | feqmptd 6249 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))) |
172 | 167, 170,
171 | 3eqtr4d 2666 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤)) |
173 | | f1oeq1 6127 |
. . . . . . 7
⊢
((((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ↔ (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
174 | 172, 173 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ↔ (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
175 | 112, 174 | mpbid 222 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
176 | 175 | ralrimivva 2971 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) |
177 | 17, 77, 100 | isffth2 16576 |
. . . 4
⊢
((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ((1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))) |
178 | 16, 176, 177 | sylanbrc 698 |
. . 3
⊢ (𝜑 → (1st
‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌)) |
179 | | df-br 4654 |
. . 3
⊢
((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ 〈(1st ‘𝑌), (2nd ‘𝑌)〉 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
180 | 178, 179 | sylib 208 |
. 2
⊢ (𝜑 → 〈(1st
‘𝑌), (2nd
‘𝑌)〉 ∈
((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
181 | 14, 180 | eqeltrd 2701 |
1
⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |