| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | idfuval.i | . 2
⊢ 𝐼 =
(idfunc‘𝐶) | 
| 2 |  | idfuval.c | . . 3
⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 3 |  | fvexd 6920 | . . . . 5
⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V) | 
| 4 |  | fveq2 6905 | . . . . . 6
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | 
| 5 |  | idfuval.b | . . . . . 6
⊢ 𝐵 = (Base‘𝐶) | 
| 6 | 4, 5 | eqtr4di 2794 | . . . . 5
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) | 
| 7 |  | simpr 484 | . . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵) | 
| 8 | 7 | reseq2d 5996 | . . . . . 6
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ 𝑏) = ( I ↾ 𝐵)) | 
| 9 | 7 | sqxpeqd 5716 | . . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵)) | 
| 10 |  | simpl 482 | . . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶) | 
| 11 | 10 | fveq2d 6909 | . . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶)) | 
| 12 |  | idfuval.h | . . . . . . . . . 10
⊢ 𝐻 = (Hom ‘𝐶) | 
| 13 | 11, 12 | eqtr4di 2794 | . . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻) | 
| 14 | 13 | fveq1d 6907 | . . . . . . . 8
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑧) = (𝐻‘𝑧)) | 
| 15 | 14 | reseq2d 5996 | . . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ( I ↾ ((Hom ‘𝑐)‘𝑧)) = ( I ↾ (𝐻‘𝑧))) | 
| 16 | 9, 15 | mpteq12dv 5232 | . . . . . 6
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))) | 
| 17 | 8, 16 | opeq12d 4880 | . . . . 5
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 〈( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))〉 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) | 
| 18 | 3, 6, 17 | csbied2 3935 | . . . 4
⊢ (𝑐 = 𝐶 → ⦋(Base‘𝑐) / 𝑏⦌〈( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))〉 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) | 
| 19 |  | df-idfu 17905 | . . . 4
⊢
idfunc = (𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌〈( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑐)‘𝑧)))〉) | 
| 20 |  | opex 5468 | . . . 4
⊢ 〈( I
↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉 ∈ V | 
| 21 | 18, 19, 20 | fvmpt 7015 | . . 3
⊢ (𝐶 ∈ Cat →
(idfunc‘𝐶) = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) | 
| 22 | 2, 21 | syl 17 | . 2
⊢ (𝜑 →
(idfunc‘𝐶) = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) | 
| 23 | 1, 22 | eqtrid 2788 | 1
⊢ (𝜑 → 𝐼 = 〈( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))〉) |