| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑢𝐵 | 
| 2 |  | nfcsb1v 3923 | . . . . 5
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵 | 
| 3 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑢𝐶 | 
| 4 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑣𝐶 | 
| 5 |  | nfcsb1v 3923 | . . . . 5
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 | 
| 6 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑦𝑢 | 
| 7 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶 | 
| 8 | 6, 7 | nfcsbw 3925 | . . . . 5
⊢
Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 | 
| 9 |  | csbeq1a 3913 | . . . . 5
⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵) | 
| 10 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶) | 
| 11 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 12 | 10, 11 | sylan9eqr 2799 | . . . . 5
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 13 | 1, 2, 3, 4, 5, 8, 9, 12 | cbvmpox 7526 | . . . 4
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 14 |  | fmpox.1 | . . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | 
| 15 |  | vex 3484 | . . . . . . . 8
⊢ 𝑢 ∈ V | 
| 16 |  | vex 3484 | . . . . . . . 8
⊢ 𝑣 ∈ V | 
| 17 | 15, 16 | op1std 8024 | . . . . . . 7
⊢ (𝑡 = 〈𝑢, 𝑣〉 → (1st ‘𝑡) = 𝑢) | 
| 18 | 17 | csbeq1d 3903 | . . . . . 6
⊢ (𝑡 = 〈𝑢, 𝑣〉 → ⦋(1st
‘𝑡) / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶) | 
| 19 | 15, 16 | op2ndd 8025 | . . . . . . . 8
⊢ (𝑡 = 〈𝑢, 𝑣〉 → (2nd ‘𝑡) = 𝑣) | 
| 20 | 19 | csbeq1d 3903 | . . . . . . 7
⊢ (𝑡 = 〈𝑢, 𝑣〉 → ⦋(2nd
‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶) | 
| 21 | 20 | csbeq2dv 3906 | . . . . . 6
⊢ (𝑡 = 〈𝑢, 𝑣〉 → ⦋𝑢 / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 22 | 18, 21 | eqtrd 2777 | . . . . 5
⊢ (𝑡 = 〈𝑢, 𝑣〉 → ⦋(1st
‘𝑡) / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 23 | 22 | mpomptx 7546 | . . . 4
⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st
‘𝑡) / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶) | 
| 24 | 13, 14, 23 | 3eqtr4i 2775 | . . 3
⊢ 𝐹 = (𝑡 ∈ ∪
𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st
‘𝑡) / 𝑥⦌⦋(2nd
‘𝑡) / 𝑦⦌𝐶) | 
| 25 | 24 | dmmptss 6261 | . 2
⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) | 
| 26 |  | nfcv 2905 | . . 3
⊢
Ⅎ𝑢({𝑥} × 𝐵) | 
| 27 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑥{𝑢} | 
| 28 | 27, 2 | nfxp 5718 | . . 3
⊢
Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) | 
| 29 |  | sneq 4636 | . . . 4
⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢}) | 
| 30 | 29, 9 | xpeq12d 5716 | . . 3
⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)) | 
| 31 | 26, 28, 30 | cbviun 5036 | . 2
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪
𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) | 
| 32 | 25, 31 | sseqtrri 4033 | 1
⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |