| Step | Hyp | Ref
| Expression |
| 1 | | ismon.s |
. 2
⊢ 𝑀 = (Mono‘𝐶) |
| 2 | | ismon.c |
. . 3
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 3 | | fvexd 6921 |
. . . . 5
⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V) |
| 4 | | fveq2 6906 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) |
| 5 | | ismon.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐶) |
| 6 | 4, 5 | eqtr4di 2795 |
. . . . 5
⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
| 7 | | fvexd 6921 |
. . . . . 6
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V) |
| 8 | | simpl 482 |
. . . . . . . 8
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶) |
| 9 | 8 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶)) |
| 10 | | ismon.h |
. . . . . . 7
⊢ 𝐻 = (Hom ‘𝐶) |
| 11 | 9, 10 | eqtr4di 2795 |
. . . . . 6
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻) |
| 12 | | simplr 769 |
. . . . . . 7
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵) |
| 13 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻) |
| 14 | 13 | oveqd 7448 |
. . . . . . . 8
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦)) |
| 15 | 13 | oveqd 7448 |
. . . . . . . . . . . 12
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑧ℎ𝑥) = (𝑧𝐻𝑥)) |
| 16 | | simpll 767 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶) |
| 17 | 16 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶)) |
| 18 | | ismon.o |
. . . . . . . . . . . . . . 15
⊢ · =
(comp‘𝐶) |
| 19 | 17, 18 | eqtr4di 2795 |
. . . . . . . . . . . . . 14
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · ) |
| 20 | 19 | oveqd 7448 |
. . . . . . . . . . . . 13
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (〈𝑧, 𝑥〉(comp‘𝑐)𝑦) = (〈𝑧, 𝑥〉 · 𝑦)) |
| 21 | 20 | oveqd 7448 |
. . . . . . . . . . . 12
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔) = (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔)) |
| 22 | 15, 21 | mpteq12dv 5233 |
. . . . . . . . . . 11
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))) |
| 23 | 22 | cnveqd 5886 |
. . . . . . . . . 10
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))) |
| 24 | 23 | funeqd 6588 |
. . . . . . . . 9
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔)))) |
| 25 | 12, 24 | raleqbidv 3346 |
. . . . . . . 8
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔)) ↔ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔)))) |
| 26 | 14, 25 | rabeqbidv 3455 |
. . . . . . 7
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))} = {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))}) |
| 27 | 12, 12, 26 | mpoeq123dv 7508 |
. . . . . 6
⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |
| 28 | 7, 11, 27 | csbied2 3936 |
. . . . 5
⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |
| 29 | 3, 6, 28 | csbied2 3936 |
. . . 4
⊢ (𝑐 = 𝐶 → ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |
| 30 | | df-mon 17774 |
. . . 4
⊢ Mono =
(𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) |
| 31 | 5 | fvexi 6920 |
. . . . 5
⊢ 𝐵 ∈ V |
| 32 | 31, 31 | mpoex 8104 |
. . . 4
⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))}) ∈ V |
| 33 | 29, 30, 32 | fvmpt 7016 |
. . 3
⊢ (𝐶 ∈ Cat →
(Mono‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |
| 34 | 2, 33 | syl 17 |
. 2
⊢ (𝜑 → (Mono‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |
| 35 | 1, 34 | eqtrid 2789 |
1
⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) |