| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7440 | . . 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 2752 | . 2
⊢ (𝑓 = 𝐹 → ((( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹)) | 
| 4 |  | oveq1 7439 | . . . 4
⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑌) = (𝑋𝐻𝑌)) | 
| 5 |  | opeq1 4872 | . . . . . . 7
⊢ (𝑥 = 𝑋 → 〈𝑥, 𝑌〉 = 〈𝑋, 𝑌〉) | 
| 6 | 5 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝑋 → (〈𝑥, 𝑌〉 · 𝑌) = (〈𝑋, 𝑌〉 · 𝑌)) | 
| 7 | 6 | oveqd 7449 | . . . . 5
⊢ (𝑥 = 𝑋 → (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓)) | 
| 8 | 7 | eqeq1d 2738 | . . . 4
⊢ (𝑥 = 𝑋 → ((( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) | 
| 9 | 4, 8 | raleqbidv 3345 | . . 3
⊢ (𝑥 = 𝑋 → (∀𝑓 ∈ (𝑥𝐻𝑌)(( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓)) | 
| 10 |  | simpl 482 | . . . . . . . 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 3082 | . . . . . . 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 4076 | . . . . 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 17721 | . . . . . 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 17719 | . . . . . . 7
⊢ (𝜑 → ∃!𝑔 ∈ (𝑌𝐻𝑌)∀𝑥 ∈ 𝐵 (∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑌𝐻𝑥)(𝑓(〈𝑌, 𝑌〉 · 𝑥)𝑔) = 𝑓)) | 
| 22 |  | riotacl2 7405 | . . . . . . 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 2840 | . . . . 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 3980 | . . . 4
⊢ (𝜑 → ( 1 ‘𝑌) ∈ {𝑔 ∈ (𝑌𝐻𝑌) ∣ ∀𝑥 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑌)(𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓}) | 
| 26 |  | oveq1 7439 | . . . . . . . 8
⊢ (𝑔 = ( 1 ‘𝑌) → (𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓)) | 
| 27 | 26 | eqeq1d 2738 | . . . . . . 7
⊢ (𝑔 = ( 1 ‘𝑌) → ((𝑔(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓 ↔ (( 1 ‘𝑌)(〈𝑥, 𝑌〉 · 𝑌)𝑓) = 𝑓)) | 
| 28 | 27 | 2ralbidv 3220 | . . . . . 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 3691 | . . . . 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 496 | . . . 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 3622 | . 2
⊢ (𝜑 → ∀𝑓 ∈ (𝑋𝐻𝑌)(( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝑓) = 𝑓) | 
| 34 |  | catlid.f | . 2
⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | 
| 35 | 3, 33, 34 | rspcdva 3622 | 1
⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) |