| Step | Hyp | Ref
| Expression |
| 1 | | mrsubffval.c |
. . . . . 6
⊢ 𝐶 = (mCN‘𝑇) |
| 2 | | mrsubffval.v |
. . . . . 6
⊢ 𝑉 = (mVR‘𝑇) |
| 3 | | mrsubffval.r |
. . . . . 6
⊢ 𝑅 = (mREx‘𝑇) |
| 4 | | mrsubffval.s |
. . . . . 6
⊢ 𝑆 = (mRSubst‘𝑇) |
| 5 | | mrsubffval.g |
. . . . . 6
⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) |
| 6 | 1, 2, 3, 4, 5 | mrsubffval 35512 |
. . . . 5
⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) |
| 7 | 6 | adantr 480 |
. . . 4
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) |
| 8 | | dmeq 5914 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) |
| 9 | | fdm 6745 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐴⟶𝑅 → dom 𝐹 = 𝐴) |
| 10 | 9 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → dom 𝐹 = 𝐴) |
| 11 | 8, 10 | sylan9eqr 2799 |
. . . . . . . . . 10
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴) |
| 12 | 11 | eleq2d 2827 |
. . . . . . . . 9
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ dom 𝑓 ↔ 𝑣 ∈ 𝐴)) |
| 13 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) |
| 14 | 13 | fveq1d 6908 |
. . . . . . . . 9
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑓‘𝑣) = (𝐹‘𝑣)) |
| 15 | 12, 14 | ifbieq1d 4550 |
. . . . . . . 8
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉) = if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) |
| 16 | 15 | mpteq2dv 5244 |
. . . . . . 7
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉))) |
| 17 | 16 | coeq1d 5872 |
. . . . . 6
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒) = ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)) |
| 18 | 17 | oveq2d 7447 |
. . . . 5
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) |
| 19 | 18 | mpteq2dv 5244 |
. . . 4
⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
| 20 | 3 | fvexi 6920 |
. . . . . 6
⊢ 𝑅 ∈ V |
| 21 | 20 | a1i 11 |
. . . . 5
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑅 ∈ V) |
| 22 | 2 | fvexi 6920 |
. . . . . 6
⊢ 𝑉 ∈ V |
| 23 | 22 | a1i 11 |
. . . . 5
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑉 ∈ V) |
| 24 | | simprl 771 |
. . . . 5
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹:𝐴⟶𝑅) |
| 25 | | simprr 773 |
. . . . 5
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐴 ⊆ 𝑉) |
| 26 | | elpm2r 8885 |
. . . . 5
⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉)) |
| 27 | 21, 23, 24, 25, 26 | syl22anc 839 |
. . . 4
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉)) |
| 28 | 20 | mptex 7243 |
. . . . 5
⊢ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) ∈ V |
| 29 | 28 | a1i 11 |
. . . 4
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) ∈ V) |
| 30 | 7, 19, 27, 29 | fvmptd 7023 |
. . 3
⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
| 31 | 30 | ex 412 |
. 2
⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) |
| 32 | | 0fv 6950 |
. . . 4
⊢
(∅‘𝐹) =
∅ |
| 33 | | fvprc 6898 |
. . . . . 6
⊢ (¬
𝑇 ∈ V →
(mRSubst‘𝑇) =
∅) |
| 34 | 4, 33 | eqtrid 2789 |
. . . . 5
⊢ (¬
𝑇 ∈ V → 𝑆 = ∅) |
| 35 | 34 | fveq1d 6908 |
. . . 4
⊢ (¬
𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹)) |
| 36 | | fvprc 6898 |
. . . . . . 7
⊢ (¬
𝑇 ∈ V →
(mREx‘𝑇) =
∅) |
| 37 | 3, 36 | eqtrid 2789 |
. . . . . 6
⊢ (¬
𝑇 ∈ V → 𝑅 = ∅) |
| 38 | 37 | mpteq1d 5237 |
. . . . 5
⊢ (¬
𝑇 ∈ V → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) = (𝑒 ∈ ∅ ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
| 39 | | mpt0 6710 |
. . . . 5
⊢ (𝑒 ∈ ∅ ↦ (𝐺 Σg
((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) = ∅ |
| 40 | 38, 39 | eqtrdi 2793 |
. . . 4
⊢ (¬
𝑇 ∈ V → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))) = ∅) |
| 41 | 32, 35, 40 | 3eqtr4a 2803 |
. . 3
⊢ (¬
𝑇 ∈ V → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
| 42 | 41 | a1d 25 |
. 2
⊢ (¬
𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) |
| 43 | 31, 42 | pm2.61i 182 |
1
⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |