Step | Hyp | Ref
| Expression |
1 | | fvex 6113 |
. . . 4
⊢
(Base‘𝑐)
∈ V |
2 | 1 | a1i 11 |
. . 3
⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V) |
3 | | fveq2 6103 |
. . . 4
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) |
4 | | iscat.b |
. . . 4
⊢ 𝐵 = (Base‘𝐶) |
5 | 3, 4 | syl6eqr 2662 |
. . 3
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | | fvex 6113 |
. . . . 5
⊢ (Hom
‘𝑐) ∈
V |
7 | 6 | a1i 11 |
. . . 4
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V) |
8 | | simpl 472 |
. . . . . 6
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶) |
9 | 8 | fveq2d 6107 |
. . . . 5
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶)) |
10 | | iscat.h |
. . . . 5
⊢ 𝐻 = (Hom ‘𝐶) |
11 | 9, 10 | syl6eqr 2662 |
. . . 4
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻) |
12 | | fvex 6113 |
. . . . . 6
⊢
(comp‘𝑐)
∈ V |
13 | 12 | a1i 11 |
. . . . 5
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) ∈ V) |
14 | | simpll 786 |
. . . . . . 7
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶) |
15 | 14 | fveq2d 6107 |
. . . . . 6
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶)) |
16 | | iscat.o |
. . . . . 6
⊢ · =
(comp‘𝐶) |
17 | 15, 16 | syl6eqr 2662 |
. . . . 5
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · ) |
18 | | simpllr 795 |
. . . . . 6
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵) |
19 | | simplr 788 |
. . . . . . . . 9
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ℎ = 𝐻) |
20 | 19 | oveqd 6566 |
. . . . . . . 8
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑥) = (𝑥𝐻𝑥)) |
21 | 19 | oveqd 6566 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥)) |
22 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · ) |
23 | 22 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑦, 𝑥〉𝑜𝑥) = (〈𝑦, 𝑥〉 · 𝑥)) |
24 | 23 | oveqd 6566 |
. . . . . . . . . . . 12
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = (𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓)) |
25 | 24 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓)) |
26 | 21, 25 | raleqbidv 3129 |
. . . . . . . . . 10
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓)) |
27 | 19 | oveqd 6566 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦)) |
28 | 22 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑥, 𝑥〉𝑜𝑦) = (〈𝑥, 𝑥〉 · 𝑦)) |
29 | 28 | oveqd 6566 |
. . . . . . . . . . . 12
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = (𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔)) |
30 | 29 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓)) |
31 | 27, 30 | raleqbidv 3129 |
. . . . . . . . . 10
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓)) |
32 | 26, 31 | anbi12d 743 |
. . . . . . . . 9
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
((∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓))) |
33 | 18, 32 | raleqbidv 3129 |
. . . . . . . 8
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓))) |
34 | 20, 33 | rexeqbidv 3130 |
. . . . . . 7
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ↔ ∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓))) |
35 | 19 | oveqd 6566 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑦ℎ𝑧) = (𝑦𝐻𝑧)) |
36 | 22 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑥, 𝑦〉𝑜𝑧) = (〈𝑥, 𝑦〉 · 𝑧)) |
37 | 36 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)) |
38 | 19 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑧) = (𝑥𝐻𝑧)) |
39 | 37, 38 | eleq12d 2682 |
. . . . . . . . . . . 12
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ↔ (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))) |
40 | 19 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑧ℎ𝑤) = (𝑧𝐻𝑤)) |
41 | 22 | oveqd 6566 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑥, 𝑦〉𝑜𝑤) = (〈𝑥, 𝑦〉 · 𝑤)) |
42 | 22 | oveqd 6566 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑦, 𝑧〉𝑜𝑤) = (〈𝑦, 𝑧〉 · 𝑤)) |
43 | 42 | oveqd 6566 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔) = (𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)) |
44 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑓 = 𝑓) |
45 | 41, 43, 44 | oveq123d 6570 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓)) |
46 | 22 | oveqd 6566 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(〈𝑥, 𝑧〉𝑜𝑤) = (〈𝑥, 𝑧〉 · 𝑤)) |
47 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑘 = 𝑘) |
48 | 46, 47, 37 | oveq123d 6570 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) |
49 | 45, 48 | eqeq12d 2625 |
. . . . . . . . . . . . . 14
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)) ↔ ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))) |
50 | 40, 49 | raleqbidv 3129 |
. . . . . . . . . . . . 13
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)) ↔ ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))) |
51 | 18, 50 | raleqbidv 3129 |
. . . . . . . . . . . 12
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)) ↔ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))) |
52 | 39, 51 | anbi12d 743 |
. . . . . . . . . . 11
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))) ↔ ((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
53 | 35, 52 | raleqbidv 3129 |
. . . . . . . . . 10
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
54 | 27, 53 | raleqbidv 3129 |
. . . . . . . . 9
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
55 | 18, 54 | raleqbidv 3129 |
. . . . . . . 8
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
56 | 18, 55 | raleqbidv 3129 |
. . . . . . 7
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))))) |
57 | 34, 56 | anbi12d 743 |
. . . . . 6
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
((∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)))) ↔ (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
58 | 18, 57 | raleqbidv 3129 |
. . . . 5
⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) →
(∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
59 | 13, 17, 58 | sbcied2 3440 |
. . . 4
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ([(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
60 | 7, 11, 59 | sbcied2 3440 |
. . 3
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ([(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
61 | 2, 5, 60 | sbcied2 3440 |
. 2
⊢ (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |
62 | | df-cat 16152 |
. 2
⊢ Cat =
{𝑐 ∣
[(Base‘𝑐) /
𝑏][(Hom
‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} |
63 | 61, 62 | elab2g 3322 |
1
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) |