| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp1 1137 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → 𝐴 ∈ 𝑉) | 
| 2 |  | simpr 484 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) | 
| 3 |  | simpl2 1193 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) | 
| 4 |  | eqid 2737 | . . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| 5 | 4 | mptfng 6707 | . . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) | 
| 6 | 3, 5 | sylibr 234 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) | 
| 7 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 | 
| 8 | 7 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ V | 
| 9 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵) | 
| 10 | 9 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑎 → (𝐵 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ V)) | 
| 11 | 8, 10 | rspc 3610 | . . . 4
⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ⦋𝑎 / 𝑥⦌𝐵 ∈ V)) | 
| 12 | 2, 6, 11 | sylc 65 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ∈ V) | 
| 13 |  | simpl3 1194 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) | 
| 14 |  | eqid 2737 | . . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | 
| 15 | 14 | mptfng 6707 | . . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) | 
| 16 | 13, 15 | sylibr 234 | . . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐶 ∈ V) | 
| 17 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 | 
| 18 | 17 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ V | 
| 19 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶) | 
| 20 | 19 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑎 → (𝐶 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ V)) | 
| 21 | 18, 20 | rspc 3610 | . . . 4
⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐶 ∈ V → ⦋𝑎 / 𝑥⦌𝐶 ∈ V)) | 
| 22 | 2, 16, 21 | sylc 65 | . . 3
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ V) | 
| 23 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑎𝐵 | 
| 24 | 23, 7, 9 | cbvmpt 5253 | . . . 4
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐵) | 
| 25 | 24 | a1i 11 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐵)) | 
| 26 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑎𝐶 | 
| 27 | 26, 17, 19 | cbvmpt 5253 | . . . 4
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐶) | 
| 28 | 27 | a1i 11 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐶)) | 
| 29 | 1, 12, 22, 25, 28 | offval2 7717 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))) | 
| 30 |  | nfcv 2905 | . . 3
⊢
Ⅎ𝑎(𝐵𝑅𝐶) | 
| 31 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑥𝑅 | 
| 32 | 7, 31, 17 | nfov 7461 | . . 3
⊢
Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶) | 
| 33 | 9, 19 | oveq12d 7449 | . . 3
⊢ (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)) | 
| 34 | 30, 32, 33 | cbvmpt 5253 | . 2
⊢ (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)) | 
| 35 | 29, 34 | eqtr4di 2795 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |