Step | Hyp | Ref
| Expression |
1 | | yoneda.y |
. . . . 5
⊢ 𝑌 = (Yon‘𝐶) |
2 | | yoneda.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐶) |
3 | | yoneda.1 |
. . . . 5
⊢ 1 =
(Id‘𝐶) |
4 | | yoneda.o |
. . . . 5
⊢ 𝑂 = (oppCat‘𝐶) |
5 | | yoneda.s |
. . . . 5
⊢ 𝑆 = (SetCat‘𝑈) |
6 | | yoneda.t |
. . . . 5
⊢ 𝑇 = (SetCat‘𝑉) |
7 | | yoneda.q |
. . . . 5
⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
8 | | yoneda.h |
. . . . 5
⊢ 𝐻 =
(HomF‘𝑄) |
9 | | yoneda.r |
. . . . 5
⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
10 | | yoneda.e |
. . . . 5
⊢ 𝐸 = (𝑂 evalF 𝑆) |
11 | | yoneda.z |
. . . . 5
⊢ 𝑍 = (𝐻 ∘func
((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉
∘func (𝑄 2ndF 𝑂))
〈,〉F (𝑄 1stF 𝑂))) |
12 | | yoneda.c |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ Cat) |
13 | | yoneda.w |
. . . . 5
⊢ (𝜑 → 𝑉 ∈ 𝑊) |
14 | | yoneda.u |
. . . . 5
⊢ (𝜑 → ran
(Homf ‘𝐶) ⊆ 𝑈) |
15 | | yoneda.v |
. . . . 5
⊢ (𝜑 → (ran
(Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
16 | | yonedalem21.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆)) |
17 | | yonedalem21.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
18 | | yonedalem4.n |
. . . . 5
⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) |
19 | | yonedalem4.p |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋)) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19 | yonedalem4a 17783 |
. . . 4
⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))) |
21 | | oveq1 7220 |
. . . . . 6
⊢ (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋)) |
22 | | oveq2 7221 |
. . . . . . . 8
⊢ (𝑦 = 𝑧 → (𝑋(2nd ‘𝐹)𝑦) = (𝑋(2nd ‘𝐹)𝑧)) |
23 | 22 | fveq1d 6719 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → ((𝑋(2nd ‘𝐹)𝑦)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑧)‘𝑔)) |
24 | 23 | fveq1d 6719 |
. . . . . 6
⊢ (𝑦 = 𝑧 → (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) |
25 | 21, 24 | mpteq12dv 5140 |
. . . . 5
⊢ (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
26 | 25 | cbvmptv 5158 |
. . . 4
⊢ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
27 | 20, 26 | eqtrdi 2794 |
. . 3
⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))) |
28 | 4, 2 | oppcbas 17222 |
. . . . . . . . . . . . 13
⊢ 𝐵 = (Base‘𝑂) |
29 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (Hom
‘𝑂) = (Hom
‘𝑂) |
30 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (Hom
‘𝑆) = (Hom
‘𝑆) |
31 | | relfunc 17368 |
. . . . . . . . . . . . . . 15
⊢ Rel
(𝑂 Func 𝑆) |
32 | | 1st2ndbr 7813 |
. . . . . . . . . . . . . . 15
⊢ ((Rel
(𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
33 | 31, 16, 32 | sylancr 590 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
34 | 33 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
35 | 17 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
36 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) |
37 | 28, 29, 30, 34, 35, 36 | funcf2 17374 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑋(2nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
38 | 37 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
39 | | simpr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) |
40 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
41 | 40, 4 | oppchom 17219 |
. . . . . . . . . . . 12
⊢ (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋) |
42 | 39, 41 | eleqtrrdi 2849 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧)) |
43 | 38, 42 | ffvelrnd 6905 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑧)‘𝑔) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
44 | 15 | unssbd 4102 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
45 | 13, 44 | ssexd 5217 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑈 ∈ V) |
46 | 45 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑈 ∈ V) |
47 | 46 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V) |
48 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(Base‘𝑆) =
(Base‘𝑆) |
49 | 28, 48, 33 | funcf1 17372 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘𝐹):𝐵⟶(Base‘𝑆)) |
50 | 5, 45 | setcbas 17584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
51 | 50 | feq3d 6532 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1st
‘𝐹):𝐵⟶𝑈 ↔ (1st ‘𝐹):𝐵⟶(Base‘𝑆))) |
52 | 49, 51 | mpbird 260 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1st
‘𝐹):𝐵⟶𝑈) |
53 | 52, 17 | ffvelrnd 6905 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1st
‘𝐹)‘𝑋) ∈ 𝑈) |
54 | 53 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈) |
55 | 52 | ffvelrnda 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st ‘𝐹)‘𝑧) ∈ 𝑈) |
56 | 55 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st ‘𝐹)‘𝑧) ∈ 𝑈) |
57 | 5, 47, 30, 54, 56 | elsetchom 17587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd ‘𝐹)𝑧)‘𝑔) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ↔ ((𝑋(2nd ‘𝐹)𝑧)‘𝑔):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑧))) |
58 | 43, 57 | mpbid 235 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑧)‘𝑔):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑧)) |
59 | 19 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st ‘𝐹)‘𝑋)) |
60 | 58, 59 | ffvelrnd 6905 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1st ‘𝐹)‘𝑧)) |
61 | 60 | fmpttd 6932 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st ‘𝐹)‘𝑧)) |
62 | 12 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐶 ∈ Cat) |
63 | 1, 2, 62, 35, 40, 36 | yon11 17772 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋)) |
64 | 63 | feq2d 6531 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st ‘𝐹)‘𝑧))) |
65 | 61, 64 | mpbird 260 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) |
66 | 1, 2, 12, 17, 4, 5, 45, 14 | yon1cl 17771 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1st
‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) |
67 | | 1st2ndbr 7813 |
. . . . . . . . . . 11
⊢ ((Rel
(𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st
‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑋))) |
68 | 31, 66, 67 | sylancr 590 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑋))) |
69 | 28, 48, 68 | funcf1 17372 |
. . . . . . . . 9
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)) |
70 | 50 | feq3d 6532 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶𝑈 ↔ (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))) |
71 | 69, 70 | mpbird 260 |
. . . . . . . 8
⊢ (𝜑 → (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶𝑈) |
72 | 71 | ffvelrnda 6904 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ∈ 𝑈) |
73 | 5, 46, 30, 72, 55 | elsetchom 17587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧))) |
74 | 65, 73 | mpbird 260 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
75 | 74 | ralrimiva 3105 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
76 | 2 | fvexi 6731 |
. . . . 5
⊢ 𝐵 ∈ V |
77 | | mptelixpg 8616 |
. . . . 5
⊢ (𝐵 ∈ V → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)))) |
78 | 76, 77 | ax-mp 5 |
. . . 4
⊢ ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
79 | 75, 78 | sylibr 237 |
. . 3
⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
80 | 27, 79 | eqeltrd 2838 |
. 2
⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
81 | 12 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝐶 ∈ Cat) |
82 | 17 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑋 ∈ 𝐵) |
83 | | simpr1 1196 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧 ∈ 𝐵) |
84 | 1, 2, 81, 82, 40, 83 | yon11 17772 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋)) |
85 | 84 | eleq2d 2823 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))) |
86 | 85 | biimpa 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) |
87 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(comp‘𝑂) =
(comp‘𝑂) |
88 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(comp‘𝑆) =
(comp‘𝑆) |
89 | 33 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
90 | 89 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹)) |
91 | 82 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑋 ∈ 𝐵) |
92 | 83 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑧 ∈ 𝐵) |
93 | | simpr2 1197 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑤 ∈ 𝐵) |
94 | 93 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑤 ∈ 𝐵) |
95 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) |
96 | 95, 41 | eleqtrrdi 2849 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧)) |
97 | | simplr3 1219 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)) |
98 | 28, 29, 87, 88, 90, 91, 92, 94, 96, 97 | funcco 17377 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑤)‘(ℎ(〈𝑋, 𝑧〉(comp‘𝑂)𝑤)𝑘)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑋(2nd ‘𝐹)𝑧)‘𝑘))) |
99 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(comp‘𝐶) =
(comp‘𝐶) |
100 | 2, 99, 4, 91, 92, 94 | oppcco 17221 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ℎ(〈𝑋, 𝑧〉(comp‘𝑂)𝑤)𝑘) = (𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ)) |
101 | 100 | fveq2d 6721 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑤)‘(ℎ(〈𝑋, 𝑧〉(comp‘𝑂)𝑤)𝑘)) = ((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))) |
102 | 45 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑈 ∈ V) |
103 | 102 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V) |
104 | 53 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈) |
105 | 55 | 3ad2antr1 1190 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘𝐹)‘𝑧) ∈ 𝑈) |
106 | 105 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st ‘𝐹)‘𝑧) ∈ 𝑈) |
107 | 52 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘𝐹):𝐵⟶𝑈) |
108 | 107, 93 | ffvelrnd 6905 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘𝐹)‘𝑤) ∈ 𝑈) |
109 | 108 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st ‘𝐹)‘𝑤) ∈ 𝑈) |
110 | 28, 29, 30, 89, 82, 83 | funcf2 17374 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑋(2nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
111 | 110 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
112 | 111, 96 | ffvelrnd 6905 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑧)‘𝑘) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧))) |
113 | 5, 103, 30, 104, 106 | elsetchom 17587 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd ‘𝐹)𝑧)‘𝑘) ∈ (((1st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ↔ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑧))) |
114 | 112, 113 | mpbid 235 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑧)‘𝑘):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑧)) |
115 | 28, 29, 30, 89, 83, 93 | funcf2 17374 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd ‘𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑤))) |
116 | | simpr3 1198 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)) |
117 | 115, 116 | ffvelrnd 6905 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∈ (((1st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑤))) |
118 | 5, 102, 30, 105, 108 | elsetchom 17587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∈ (((1st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑤)) ↔ ((𝑧(2nd ‘𝐹)𝑤)‘ℎ):((1st ‘𝐹)‘𝑧)⟶((1st ‘𝐹)‘𝑤))) |
119 | 117, 118 | mpbid 235 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘𝐹)𝑤)‘ℎ):((1st ‘𝐹)‘𝑧)⟶((1st ‘𝐹)‘𝑤)) |
120 | 119 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑧(2nd ‘𝐹)𝑤)‘ℎ):((1st ‘𝐹)‘𝑧)⟶((1st ‘𝐹)‘𝑤)) |
121 | 5, 103, 88, 104, 106, 109, 114, 120 | setcco 17589 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑋(2nd ‘𝐹)𝑧)‘𝑘)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘))) |
122 | 98, 101, 121 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘))) |
123 | 122 | fveq1d 6719 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))‘𝐴) = ((((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘))‘𝐴)) |
124 | 19 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st ‘𝐹)‘𝑋)) |
125 | | fvco3 6810 |
. . . . . . . . . 10
⊢ ((((𝑋(2nd ‘𝐹)𝑧)‘𝑘):((1st ‘𝐹)‘𝑋)⟶((1st ‘𝐹)‘𝑧) ∧ 𝐴 ∈ ((1st ‘𝐹)‘𝑋)) → ((((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2nd ‘𝐹)𝑧)‘𝑘)‘𝐴))) |
126 | 114, 124,
125 | syl2anc 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2nd ‘𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2nd ‘𝐹)𝑧)‘𝑘)‘𝐴))) |
127 | 123, 126 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))‘𝐴) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2nd ‘𝐹)𝑧)‘𝑘)‘𝐴))) |
128 | 81 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat) |
129 | 40, 4 | oppchom 17219 |
. . . . . . . . . . . 12
⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧) |
130 | 97, 129 | eleqtrdi 2848 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ℎ ∈ (𝑤(Hom ‘𝐶)𝑧)) |
131 | 1, 2, 128, 91, 40, 92, 99, 94, 130, 95 | yon12 17773 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘) = (𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ)) |
132 | 131 | fveq2d 6721 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))) |
133 | 13 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑉 ∈ 𝑊) |
134 | 14 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ran (Homf
‘𝐶) ⊆ 𝑈) |
135 | 15 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ran (Homf
‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
136 | 16 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐹 ∈ (𝑂 Func 𝑆)) |
137 | 2, 40, 99, 128, 94, 92, 91, 130, 95 | catcocl 17188 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ) ∈ (𝑤(Hom ‘𝐶)𝑋)) |
138 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 128, 133, 134, 135, 136, 91, 18, 124, 94, 137 | yonedalem4b 17784 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ)) = (((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))‘𝐴)) |
139 | 132, 138 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑋(2nd ‘𝐹)𝑤)‘(𝑘(〈𝑤, 𝑧〉(comp‘𝐶)𝑋)ℎ))‘𝐴)) |
140 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 128, 133, 134, 135, 136, 91, 18, 124, 92, 95 | yonedalem4b 17784 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2nd ‘𝐹)𝑧)‘𝑘)‘𝐴)) |
141 | 140 | fveq2d 6721 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2nd ‘𝐹)𝑧)‘𝑘)‘𝐴))) |
142 | 127, 139,
141 | 3eqtr4d 2787 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))) |
143 | 86, 142 | syldan 594 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))) |
144 | 143 | mpteq2dva 5150 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘))) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))) |
145 | | fveq2 6717 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤)) |
146 | | fveq2 6717 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) = ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)) |
147 | | fveq2 6717 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → ((1st ‘𝐹)‘𝑧) = ((1st ‘𝐹)‘𝑤)) |
148 | 145, 146,
147 | feq123d 6534 |
. . . . . . 7
⊢ (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)⟶((1st ‘𝐹)‘𝑤))) |
149 | 27 | fveq1d 6719 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧)) |
150 | | ovex 7246 |
. . . . . . . . . . . . . 14
⊢ (𝑧(Hom ‘𝐶)𝑋) ∈ V |
151 | 150 | mptex 7039 |
. . . . . . . . . . . . 13
⊢ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V |
152 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
153 | 152 | fvmpt2 6829 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝐵 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V) → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
154 | 151, 153 | mpan2 691 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝐵 → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
155 | 149, 154 | sylan9eq 2798 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) |
156 | 155 | feq1d 6530 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧))) |
157 | 65, 156 | mpbird 260 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) |
158 | 157 | ralrimiva 3105 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) |
159 | 158 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∀𝑧 ∈ 𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) |
160 | 148, 159,
93 | rspcdva 3539 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)⟶((1st ‘𝐹)‘𝑤)) |
161 | 68 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st
‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st
‘𝑌)‘𝑋))) |
162 | 28, 29, 30, 161, 83, 93 | funcf2 17374 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤))) |
163 | 162, 116 | ffvelrnd 6905 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤))) |
164 | 72 | 3ad2antr1 1190 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ∈ 𝑈) |
165 | 71 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st
‘((1st ‘𝑌)‘𝑋)):𝐵⟶𝑈) |
166 | 165, 93 | ffvelrnd 6905 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑤) ∈ 𝑈) |
167 | 5, 102, 30, 164, 166 | elsetchom 17587 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ) ∈ (((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)) ↔ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑤))) |
168 | 163, 167 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑤)) |
169 | | fcompt 6948 |
. . . . . 6
⊢
(((((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)⟶((1st ‘𝐹)‘𝑤) ∧ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st
‘((1st ‘𝑌)‘𝑋))‘𝑤)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)))) |
170 | 160, 168,
169 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)))) |
171 | 157 | 3ad2antr1 1190 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) |
172 | | fcompt 6948 |
. . . . . 6
⊢ ((((𝑧(2nd ‘𝐹)𝑤)‘ℎ):((1st ‘𝐹)‘𝑧)⟶((1st ‘𝐹)‘𝑤) ∧ (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)⟶((1st ‘𝐹)‘𝑧)) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))) |
173 | 119, 171,
172 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))) |
174 | 144, 170,
173 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧))) |
175 | 5, 102, 88, 164, 166, 108, 168, 160 | setcco 17589 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ))) |
176 | 5, 102, 88, 164, 105, 108, 171, 119 | setcco 17589 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧))) |
177 | 174, 175,
176 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧))) |
178 | 177 | ralrimivvva 3113 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧))) |
179 | | eqid 2737 |
. . 3
⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆) |
180 | 179, 28, 29, 30, 88, 66, 16 | isnat2 17455 |
. 2
⊢ (𝜑 → (((𝐹𝑁𝑋)‘𝐴) ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↔ (((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘𝐹)‘𝑧)) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝑌)‘𝑋))‘𝑤)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))((𝑧(2nd ‘((1st
‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2nd ‘𝐹)𝑤)‘ℎ)(〈((1st
‘((1st ‘𝑌)‘𝑋))‘𝑧), ((1st ‘𝐹)‘𝑧)〉(comp‘𝑆)((1st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧))))) |
181 | 80, 178, 180 | mpbir2and 713 |
1
⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) |