| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | catidd.2 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) | 
| 2 | 1 | ex 412 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥)) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓)) | 
| 3 |  | catidd.b | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | 
| 4 | 3 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐶))) | 
| 5 | 3 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐶))) | 
| 6 |  | catidd.h | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | 
| 7 | 6 | oveqd 7449 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑦𝐻𝑥) = (𝑦(Hom ‘𝐶)𝑥)) | 
| 8 | 7 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑓 ∈ (𝑦𝐻𝑥) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) | 
| 9 | 4, 5, 8 | 3anbi123d 1437 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)))) | 
| 10 |  | catidd.o | . . . . . . . . . . . . 13
⊢ (𝜑 → · = (comp‘𝐶)) | 
| 11 | 10 | oveqd 7449 | . . . . . . . . . . . 12
⊢ (𝜑 → (〈𝑦, 𝑥〉 · 𝑥) = (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)) | 
| 12 | 11 | oveqd 7449 | . . . . . . . . . . 11
⊢ (𝜑 → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓)) | 
| 13 | 12 | eqeq1d 2738 | . . . . . . . . . 10
⊢ (𝜑 → (( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ↔ ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓)) | 
| 14 | 2, 9, 13 | 3imtr3d 293 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓)) | 
| 15 | 14 | 3expd 1353 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥) → ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓)))) | 
| 16 | 15 | imp41 425 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓) | 
| 17 | 16 | ralrimiva 3145 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓) | 
| 18 |  | catidd.3 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) | 
| 19 | 18 | ex 412 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓)) | 
| 20 | 6 | oveqd 7449 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) | 
| 21 | 20 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) | 
| 22 | 4, 5, 21 | 3anbi123d 1437 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))) | 
| 23 | 10 | oveqd 7449 | . . . . . . . . . . . 12
⊢ (𝜑 → (〈𝑥, 𝑥〉 · 𝑦) = (〈𝑥, 𝑥〉(comp‘𝐶)𝑦)) | 
| 24 | 23 | oveqd 7449 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 )) | 
| 25 | 24 | eqeq1d 2738 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓 ↔ (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 26 | 19, 22, 25 | 3imtr3d 293 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 27 | 26 | 3expd 1353 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) → (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)))) | 
| 28 | 27 | imp41 425 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓) | 
| 29 | 28 | ralrimiva 3145 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓) | 
| 30 | 17, 29 | jca 511 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 31 | 30 | ralrimiva 3145 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 32 |  | catidd.1 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) | 
| 33 | 32 | ex 412 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐵 → 1 ∈ (𝑥𝐻𝑥))) | 
| 34 | 6 | oveqd 7449 | . . . . . . . 8
⊢ (𝜑 → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥)) | 
| 35 | 34 | eleq2d 2826 | . . . . . . 7
⊢ (𝜑 → ( 1 ∈ (𝑥𝐻𝑥) ↔ 1 ∈ (𝑥(Hom ‘𝐶)𝑥))) | 
| 36 | 33, 4, 35 | 3imtr3d 293 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥))) | 
| 37 | 36 | imp 406 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥)) | 
| 38 |  | eqid 2736 | . . . . . 6
⊢
(Base‘𝐶) =
(Base‘𝐶) | 
| 39 |  | eqid 2736 | . . . . . 6
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) | 
| 40 |  | eqid 2736 | . . . . . 6
⊢
(comp‘𝐶) =
(comp‘𝐶) | 
| 41 |  | catidd.c | . . . . . . 7
⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 42 | 41 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat) | 
| 43 |  | simpr 484 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶)) | 
| 44 | 38, 39, 40, 42, 43 | catideu 17719 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) | 
| 45 |  | oveq1 7439 | . . . . . . . . . 10
⊢ (𝑔 = 1 → (𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓)) | 
| 46 | 45 | eqeq1d 2738 | . . . . . . . . 9
⊢ (𝑔 = 1 → ((𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓)) | 
| 47 | 46 | ralbidv 3177 | . . . . . . . 8
⊢ (𝑔 = 1 → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓)) | 
| 48 |  | oveq2 7440 | . . . . . . . . . 10
⊢ (𝑔 = 1 → (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 )) | 
| 49 | 48 | eqeq1d 2738 | . . . . . . . . 9
⊢ (𝑔 = 1 → ((𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 50 | 49 | ralbidv 3177 | . . . . . . . 8
⊢ (𝑔 = 1 → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓)) | 
| 51 | 47, 50 | anbi12d 632 | . . . . . . 7
⊢ (𝑔 = 1 → ((∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓))) | 
| 52 | 51 | ralbidv 3177 | . . . . . 6
⊢ (𝑔 = 1 → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓))) | 
| 53 | 52 | riota2 7414 | . . . . 5
⊢ (( 1 ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 )) | 
| 54 | 37, 44, 53 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 )) | 
| 55 | 31, 54 | mpbid 232 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 ) | 
| 56 | 55 | mpteq2dva 5241 | . 2
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) = (𝑥 ∈ (Base‘𝐶) ↦ 1 )) | 
| 57 |  | eqid 2736 | . . 3
⊢
(Id‘𝐶) =
(Id‘𝐶) | 
| 58 | 38, 39, 40, 41, 57 | cidfval 17720 | . 2
⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ (Base‘𝐶) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)))) | 
| 59 | 3 | mpteq1d 5236 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 1 ) = (𝑥 ∈ (Base‘𝐶) ↦ 1 )) | 
| 60 | 56, 58, 59 | 3eqtr4d 2786 | 1
⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 )) |