Step | Hyp | Ref
| Expression |
1 | | oveq2 7275 |
. . 3
⊢ (𝑓 = 𝐹 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹)) |
2 | | id 22 |
. . 3
⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) |
3 | 1, 2 | eqeq12d 2754 |
. 2
⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹)) |
4 | | oveq1 7274 |
. . . 4
⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌)) |
5 | | opeq1 4804 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → 〈𝑥, 𝑌〉 = 〈𝑋, 𝑌〉) |
6 | 5 | oveq1d 7282 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (〈𝑥, 𝑌〉 · 𝑌) = (〈𝑋, 𝑌〉 · 𝑌)) |
7 | 6 | oveqd 7284 |
. . . . 5
⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓)) |
8 | 7 | eqeq1d 2740 |
. . . 4
⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
9 | 4, 8 | raleqbidv 3334 |
. . 3
⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
10 | | simpl 483 |
. . . . . . . 8
⊢
((∀𝑓 ∈
(𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓) → ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
11 | 10 | ralimi 3087 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓) → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
12 | 11 | a1i 11 |
. . . . . 6
⊢ (𝑔 ∈ (𝑌𝐻𝑌) → (∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓) → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
13 | 12 | ss2rabi 4009 |
. . . . 5
⊢ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)} ⊆ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓} |
14 | | catidcl.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐶) |
15 | | catidcl.h |
. . . . . . 7
⊢ 𝐻 = (Hom ‘𝐶) |
16 | | catlid.o |
. . . . . . 7
⊢ · =
(comp‘𝐶) |
17 | | catidcl.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ Cat) |
18 | | catidcl.i |
. . . . . . 7
⊢ 1 =
(Id‘𝐶) |
19 | | catlid.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
20 | 14, 15, 16, 17, 18, 19 | cidval 17396 |
. . . . . 6
⊢ (𝜑 → ( 1 ‘𝑌) = (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓))) |
21 | 14, 15, 16, 17, 19 | catideu 17394 |
. . . . . . 7
⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)) |
22 | | riotacl2 7241 |
. . . . . . 7
⊢
(∃!𝑔 ∈
(𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓) → (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)}) |
23 | 21, 22 | syl 17 |
. . . . . 6
⊢ (𝜑 → (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)}) |
24 | 20, 23 | eqeltrd 2839 |
. . . . 5
⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)}) |
25 | 13, 24 | sselid 3918 |
. . . 4
⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓}) |
26 | | oveq1 7274 |
. . . . . . . 8
⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓)) |
27 | 26 | eqeq1d 2740 |
. . . . . . 7
⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
28 | 27 | 2ralbidv 3123 |
. . . . . 6
⊢ (𝑔 = ( 1 ‘𝑌) → (∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
29 | 28 | elrab 3623 |
. . . . 5
⊢ (( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓} ↔ (( 1 ‘𝑌) ∈ (𝑌𝐻𝑌) ∧ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
30 | 29 | simprbi 497 |
. . . 4
⊢ (( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓} → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
31 | 25, 30 | syl 17 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
32 | | catidcl.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
33 | 9, 31, 32 | rspcdva 3561 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
34 | | catlid.f |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
35 | 3, 33, 34 | rspcdva 3561 |
1
⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) |