Step | Hyp | Ref
| Expression |
1 | | catccatid.b |
. . 3
⊢ 𝐵 = (Base‘𝐶) |
2 | 1 | a1i 11 |
. 2
⊢ (𝑈 ∈ 𝑉 → 𝐵 = (Base‘𝐶)) |
3 | | eqidd 2623 |
. 2
⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶)) |
4 | | eqidd 2623 |
. 2
⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶)) |
5 | | catccatid.c |
. . . 4
⊢ 𝐶 = (CatCat‘𝑈) |
6 | | fvex 6201 |
. . . 4
⊢
(CatCat‘𝑈)
∈ V |
7 | 5, 6 | eqeltri 2697 |
. . 3
⊢ 𝐶 ∈ V |
8 | 7 | a1i 11 |
. 2
⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
9 | | biid 251 |
. 2
⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) |
10 | | id 22 |
. . . . . . 7
⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉) |
11 | 5, 1, 10 | catcbas 16747 |
. . . . . 6
⊢ (𝑈 ∈ 𝑉 → 𝐵 = (𝑈 ∩ Cat)) |
12 | | inss2 3834 |
. . . . . 6
⊢ (𝑈 ∩ Cat) ⊆
Cat |
13 | 11, 12 | syl6eqss 3655 |
. . . . 5
⊢ (𝑈 ∈ 𝑉 → 𝐵 ⊆ Cat) |
14 | 13 | sselda 3603 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Cat) |
15 | | eqid 2622 |
. . . . 5
⊢
(idfunc‘𝑥) = (idfunc‘𝑥) |
16 | 15 | idfucl 16541 |
. . . 4
⊢ (𝑥 ∈ Cat →
(idfunc‘𝑥) ∈ (𝑥 Func 𝑥)) |
17 | 14, 16 | syl 17 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) →
(idfunc‘𝑥) ∈ (𝑥 Func 𝑥)) |
18 | | simpl 473 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
19 | | eqid 2622 |
. . . 4
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
20 | | simpr 477 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
21 | 5, 1, 18, 19, 20, 20 | catchom 16749 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 Func 𝑥)) |
22 | 17, 21 | eleqtrrd 2704 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) →
(idfunc‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) |
23 | | simpl 473 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉) |
24 | | eqid 2622 |
. . . 4
⊢
(comp‘𝐶) =
(comp‘𝐶) |
25 | | simpr1l 1118 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝐵) |
26 | | simpr1r 1119 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝐵) |
27 | | simpr31 1151 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥)) |
28 | 5, 1, 23, 19, 25, 26 | catchom 16749 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 Func 𝑥)) |
29 | 27, 28 | eleqtrd 2703 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 Func 𝑥)) |
30 | 26, 17 | syldan 487 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) →
(idfunc‘𝑥) ∈ (𝑥 Func 𝑥)) |
31 | 5, 1, 23, 24, 25, 26, 26, 29, 30 | catcco 16751 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) →
((idfunc‘𝑥)(〈𝑤, 𝑥〉(comp‘𝐶)𝑥)𝑓) = ((idfunc‘𝑥) ∘func
𝑓)) |
32 | 29, 15 | cofulid 16550 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) →
((idfunc‘𝑥) ∘func 𝑓) = 𝑓) |
33 | 31, 32 | eqtrd 2656 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) →
((idfunc‘𝑥)(〈𝑤, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓) |
34 | | simpr2l 1120 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝐵) |
35 | | simpr32 1152 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
36 | 5, 1, 23, 19, 26, 34 | catchom 16749 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦)) |
37 | 35, 36 | eleqtrd 2703 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 Func 𝑦)) |
38 | 5, 1, 23, 24, 26, 26, 34, 30, 37 | catcco 16751 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)(idfunc‘𝑥)) = (𝑔 ∘func
(idfunc‘𝑥))) |
39 | 37, 15 | cofurid 16551 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘func
(idfunc‘𝑥)) = 𝑔) |
40 | 38, 39 | eqtrd 2656 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)(idfunc‘𝑥)) = 𝑔) |
41 | 29, 37 | cofucl 16548 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘func 𝑓) ∈ (𝑤 Func 𝑦)) |
42 | 5, 1, 23, 24, 25, 26, 34, 29, 37 | catcco 16751 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(〈𝑤, 𝑥〉(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘func 𝑓)) |
43 | 5, 1, 23, 19, 25, 34 | catchom 16749 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 Func 𝑦)) |
44 | 41, 42, 43 | 3eltr4d 2716 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(〈𝑤, 𝑥〉(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦)) |
45 | | simpr33 1153 |
. . . . . 6
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) |
46 | | simpr2r 1121 |
. . . . . . 7
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝐵) |
47 | 5, 1, 23, 19, 34, 46 | catchom 16749 |
. . . . . 6
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 Func 𝑧)) |
48 | 45, 47 | eleqtrd 2703 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦 Func 𝑧)) |
49 | 29, 37, 48 | cofuass 16549 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔) ∘func
𝑓) = (ℎ ∘func (𝑔 ∘func
𝑓))) |
50 | 37, 48 | cofucl 16548 |
. . . . 5
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘func 𝑔) ∈ (𝑥 Func 𝑧)) |
51 | 5, 1, 23, 24, 25, 26, 46, 29, 50 | catcco 16751 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔)(〈𝑤, 𝑥〉(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘func 𝑔) ∘func
𝑓)) |
52 | 5, 1, 23, 24, 25, 34, 46, 41, 48 | catcco 16751 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(〈𝑤, 𝑦〉(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓)) = (ℎ ∘func (𝑔 ∘func
𝑓))) |
53 | 49, 51, 52 | 3eqtr4d 2666 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔)(〈𝑤, 𝑥〉(comp‘𝐶)𝑧)𝑓) = (ℎ(〈𝑤, 𝑦〉(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓))) |
54 | 5, 1, 23, 24, 26, 34, 46, 37, 48 | catcco 16751 |
. . . 4
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑔) = (ℎ ∘func 𝑔)) |
55 | 54 | oveq1d 6665 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑔)(〈𝑤, 𝑥〉(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘func 𝑔)(〈𝑤, 𝑥〉(comp‘𝐶)𝑧)𝑓)) |
56 | 42 | oveq2d 6666 |
. . 3
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(〈𝑤, 𝑦〉(comp‘𝐶)𝑧)(𝑔(〈𝑤, 𝑥〉(comp‘𝐶)𝑦)𝑓)) = (ℎ(〈𝑤, 𝑦〉(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓))) |
57 | 53, 55, 56 | 3eqtr4d 2666 |
. 2
⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑔)(〈𝑤, 𝑥〉(comp‘𝐶)𝑧)𝑓) = (ℎ(〈𝑤, 𝑦〉(comp‘𝐶)𝑧)(𝑔(〈𝑤, 𝑥〉(comp‘𝐶)𝑦)𝑓))) |
58 | 2, 3, 4, 8, 9, 22,
33, 40, 44, 57 | iscatd2 16342 |
1
⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦
(idfunc‘𝑥)))) |