Step | Hyp | Ref
| Expression |
1 | | catlid.f |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
2 | | catlid.y |
. . 3
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
3 | | simpr 477 |
. . . . . . . 8
⊢
((∀𝑓 ∈
(𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) |
4 | 3 | ralimi 2952 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) |
5 | 4 | a1i 11 |
. . . . . 6
⊢ (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) |
6 | 5 | ss2rabi 3684 |
. . . . 5
⊢ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓} |
7 | | catidcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐶) |
8 | | catidcl.h |
. . . . . . 7
⊢ 𝐻 = (Hom ‘𝐶) |
9 | | catlid.o |
. . . . . . 7
⊢ · =
(comp‘𝐶) |
10 | | catidcl.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ Cat) |
11 | | catidcl.i |
. . . . . . 7
⊢ 1 =
(Id‘𝐶) |
12 | | catidcl.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
13 | 7, 8, 9, 10, 11, 12 | cidval 16338 |
. . . . . 6
⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓))) |
14 | 7, 8, 9, 10, 12 | catideu 16336 |
. . . . . . 7
⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) |
15 | | riotacl2 6624 |
. . . . . . 7
⊢
(∃!𝑔 ∈
(𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)}) |
16 | 14, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)}) |
17 | 13, 16 | eqeltrd 2701 |
. . . . 5
⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)}) |
18 | 6, 17 | sseldi 3601 |
. . . 4
⊢ (𝜑 → ( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓}) |
19 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑔 = ( 1 ‘𝑋) → (𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = (𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋))) |
20 | 19 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑔 = ( 1 ‘𝑋) → ((𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓 ↔ (𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓)) |
21 | 20 | 2ralbidv 2989 |
. . . . . 6
⊢ (𝑔 = ( 1 ‘𝑋) → (∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓 ↔ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓)) |
22 | 21 | elrab 3363 |
. . . . 5
⊢ (( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓} ↔ (( 1 ‘𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓)) |
23 | 22 | simprbi 480 |
. . . 4
⊢ (( 1 ‘𝑋) ∈ {𝑔 ∈ (𝑋𝐻𝑋) ∣ ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓} → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓) |
24 | 18, 23 | syl 17 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓) |
25 | | oveq2 6658 |
. . . . 5
⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌)) |
26 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑦 = 𝑌 → (〈𝑋, 𝑋〉 · 𝑦) = (〈𝑋, 𝑋〉 · 𝑌)) |
27 | 26 | oveqd 6667 |
. . . . . 6
⊢ (𝑦 = 𝑌 → (𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = (𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋))) |
28 | 27 | eqeq1d 2624 |
. . . . 5
⊢ (𝑦 = 𝑌 → ((𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ (𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓)) |
29 | 25, 28 | raleqbidv 3152 |
. . . 4
⊢ (𝑦 = 𝑌 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓)) |
30 | 29 | rspcv 3305 |
. . 3
⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)( 1 ‘𝑋)) = 𝑓 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓)) |
31 | 2, 24, 30 | sylc 65 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓) |
32 | | oveq1 6657 |
. . . 4
⊢ (𝑓 = 𝐹 → (𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋))) |
33 | | id 22 |
. . . 4
⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) |
34 | 32, 33 | eqeq12d 2637 |
. . 3
⊢ (𝑓 = 𝐹 → ((𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓 ↔ (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹)) |
35 | 34 | rspcv 3305 |
. 2
⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (∀𝑓 ∈ (𝑋𝐻𝑌)(𝑓(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝑓 → (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹)) |
36 | 1, 31, 35 | sylc 65 |
1
⊢ (𝜑 → (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹) |