Step | Hyp | Ref
| Expression |
1 | | catlid.f |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
2 | | catidcl.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
3 | | simpl 472 |
. . . . . . . 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) |
4 | 3 | ralimi 2936 |
. . . . . . 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) |
5 | 4 | 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)) |
6 | 5 | ss2rabi 3647 |
. . . . 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} |
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 | | catlid.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
13 | 7, 8, 9, 10, 11, 12 | cidval 16161 |
. . . . . 6
⊢ (𝜑 → ( 1 ‘𝑌) = (℩𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓))) |
14 | 7, 8, 9, 10, 12 | catideu 16159 |
. . . . . . 7
⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)) |
15 | | riotacl2 6524 |
. . . . . . 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)}) |
16 | 14, 15 | 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)}) |
17 | 13, 16 | eqeltrd 2688 |
. . . . 5
⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)}) |
18 | 6, 17 | sseldi 3566 |
. . . 4
⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓}) |
19 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓)) |
20 | 19 | eqeq1d 2612 |
. . . . . . 7
⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
21 | 20 | 2ralbidv 2972 |
. . . . . 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)) |
22 | 21 | elrab 3331 |
. . . . 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)) |
23 | 22 | simprbi 479 |
. . . 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) |
24 | 18, 23 | syl 17 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
25 | | oveq1 6556 |
. . . . 5
⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌)) |
26 | | opeq1 4340 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → 〈𝑥, 𝑌〉 = 〈𝑋, 𝑌〉) |
27 | 26 | oveq1d 6564 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → (〈𝑥, 𝑌〉 · 𝑌) = (〈𝑋, 𝑌〉 · 𝑌)) |
28 | 27 | oveqd 6566 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓)) |
29 | 28 | eqeq1d 2612 |
. . . . 5
⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
30 | 25, 29 | raleqbidv 3129 |
. . . 4
⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
31 | 30 | rspcv 3278 |
. . 3
⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) |
32 | 2, 24, 31 | sylc 63 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓) |
33 | | oveq2 6557 |
. . . 4
⊢ (𝑓 = 𝐹 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹)) |
34 | | id 22 |
. . . 4
⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) |
35 | 33, 34 | eqeq12d 2625 |
. . 3
⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹)) |
36 | 35 | rspcv 3278 |
. 2
⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹)) |
37 | 1, 32, 36 | sylc 63 |
1
⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) |