| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | issect.s | . 2
⊢ 𝑆 = (Sect‘𝐶) | 
| 2 |  | issect.c | . . 3
⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 3 |  | fveq2 6906 | . . . . . 6
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | 
| 4 |  | issect.b | . . . . . 6
⊢ 𝐵 = (Base‘𝐶) | 
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) | 
| 6 |  | fvexd 6921 | . . . . . . 7
⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V) | 
| 7 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | 
| 8 |  | issect.h | . . . . . . . 8
⊢ 𝐻 = (Hom ‘𝐶) | 
| 9 | 7, 8 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) | 
| 10 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻) | 
| 11 | 10 | oveqd 7448 | . . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦)) | 
| 12 | 11 | eleq2d 2827 | . . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑓 ∈ (𝑥ℎ𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦))) | 
| 13 | 10 | oveqd 7448 | . . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥)) | 
| 14 | 13 | eleq2d 2827 | . . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑦ℎ𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥))) | 
| 15 | 12, 14 | anbi12d 632 | . . . . . . . 8
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)))) | 
| 16 |  | simpl 482 | . . . . . . . . . . . . 13
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶) | 
| 17 | 16 | fveq2d 6910 | . . . . . . . . . . . 12
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶)) | 
| 18 |  | issect.o | . . . . . . . . . . . 12
⊢  · =
(comp‘𝐶) | 
| 19 | 17, 18 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (comp‘𝑐) = · ) | 
| 20 | 19 | oveqd 7448 | . . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (〈𝑥, 𝑦〉(comp‘𝑐)𝑥) = (〈𝑥, 𝑦〉 · 𝑥)) | 
| 21 | 20 | oveqd 7448 | . . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓)) | 
| 22 | 16 | fveq2d 6910 | . . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = (Id‘𝐶)) | 
| 23 |  | issect.i | . . . . . . . . . . 11
⊢  1 =
(Id‘𝐶) | 
| 24 | 22, 23 | eqtr4di 2795 | . . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (Id‘𝑐) = 1 ) | 
| 25 | 24 | fveq1d 6908 | . . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥)) | 
| 26 | 21, 25 | eqeq12d 2753 | . . . . . . . 8
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))) | 
| 27 | 15, 26 | anbi12d 632 | . . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥)))) | 
| 28 | 6, 9, 27 | sbcied2 3833 | . . . . . 6
⊢ (𝑐 = 𝐶 → ([(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥)))) | 
| 29 | 28 | opabbidv 5209 | . . . . 5
⊢ (𝑐 = 𝐶 → {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))} = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))}) | 
| 30 | 5, 5, 29 | mpoeq123dv 7508 | . . . 4
⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) | 
| 31 |  | df-sect 17791 | . . . 4
⊢ Sect =
(𝑐 ∈ Cat ↦
(𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))})) | 
| 32 | 4 | fvexi 6920 | . . . . 5
⊢ 𝐵 ∈ V | 
| 33 | 32, 32 | mpoex 8104 | . . . 4
⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))}) ∈ V | 
| 34 | 30, 31, 33 | fvmpt 7016 | . . 3
⊢ (𝐶 ∈ Cat →
(Sect‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) | 
| 35 | 2, 34 | syl 17 | . 2
⊢ (𝜑 → (Sect‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) | 
| 36 | 1, 35 | eqtrid 2789 | 1
⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) |