| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfcv 2904 | . . . . 5
⊢
Ⅎ𝑢𝐴 | 
| 2 |  | nfcsb1v 3922 | . . . . 5
⊢
Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴 | 
| 3 |  | nfcv 2904 | . . . . 5
⊢
Ⅎ𝑢𝐶 | 
| 4 |  | nfcv 2904 | . . . . 5
⊢
Ⅎ𝑣𝐶 | 
| 5 |  | nfcv 2904 | . . . . . 6
⊢
Ⅎ𝑥𝑢 | 
| 6 |  | nfcsb1v 3922 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶 | 
| 7 | 5, 6 | nfcsbw 3924 | . . . . 5
⊢
Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶 | 
| 8 |  | nfcsb1v 3922 | . . . . 5
⊢
Ⅎ𝑦⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶 | 
| 9 |  | csbeq1a 3912 | . . . . 5
⊢ (𝑦 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑦⦌𝐴) | 
| 10 |  | csbeq1a 3912 | . . . . . 6
⊢ (𝑥 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑥⦌𝐶) | 
| 11 |  | csbeq1a 3912 | . . . . . 6
⊢ (𝑦 = 𝑢 → ⦋𝑣 / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 12 | 10, 11 | sylan9eqr 2798 | . . . . 5
⊢ ((𝑦 = 𝑢 ∧ 𝑥 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 13 | 1, 2, 3, 4, 7, 8, 9, 12 | cbvmpox2 48257 | . . . 4
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑣 ∈ ⦋𝑢 / 𝑦⦌𝐴, 𝑢 ∈ 𝐵 ↦ ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 14 |  | dmmpossx2.1 | . . . 4
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | 
| 15 |  | vex 3483 | . . . . . . . 8
⊢ 𝑣 ∈ V | 
| 16 |  | vex 3483 | . . . . . . . 8
⊢ 𝑢 ∈ V | 
| 17 | 15, 16 | op2ndd 8026 | . . . . . . 7
⊢ (𝑡 = 〈𝑣, 𝑢〉 → (2nd ‘𝑡) = 𝑢) | 
| 18 | 17 | csbeq1d 3902 | . . . . . 6
⊢ (𝑡 = 〈𝑣, 𝑢〉 → ⦋(2nd
‘𝑡) / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶) | 
| 19 | 15, 16 | op1std 8025 | . . . . . . . 8
⊢ (𝑡 = 〈𝑣, 𝑢〉 → (1st ‘𝑡) = 𝑣) | 
| 20 | 19 | csbeq1d 3902 | . . . . . . 7
⊢ (𝑡 = 〈𝑣, 𝑢〉 → ⦋(1st
‘𝑡) / 𝑥⦌𝐶 = ⦋𝑣 / 𝑥⦌𝐶) | 
| 21 | 20 | csbeq2dv 3905 | . . . . . 6
⊢ (𝑡 = 〈𝑣, 𝑢〉 → ⦋𝑢 / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 22 | 18, 21 | eqtrd 2776 | . . . . 5
⊢ (𝑡 = 〈𝑣, 𝑢〉 → ⦋(2nd
‘𝑡) / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 23 | 22 | mpomptx2 48256 | . . . 4
⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd
‘𝑡) / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶) = (𝑣 ∈ ⦋𝑢 / 𝑦⦌𝐴, 𝑢 ∈ 𝐵 ↦ ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶) | 
| 24 | 13, 14, 23 | 3eqtr4i 2774 | . . 3
⊢ 𝐹 = (𝑡 ∈ ∪
𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd
‘𝑡) / 𝑦⦌⦋(1st
‘𝑡) / 𝑥⦌𝐶) | 
| 25 | 24 | dmmptss 6260 | . 2
⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) | 
| 26 |  | nfcv 2904 | . . 3
⊢
Ⅎ𝑢(𝐴 × {𝑦}) | 
| 27 |  | nfcv 2904 | . . . 4
⊢
Ⅎ𝑦{𝑢} | 
| 28 | 2, 27 | nfxp 5717 | . . 3
⊢
Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) | 
| 29 |  | sneq 4635 | . . . 4
⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢}) | 
| 30 | 9, 29 | xpeq12d 5715 | . . 3
⊢ (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})) | 
| 31 | 26, 28, 30 | cbviun 5035 | . 2
⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = ∪
𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) | 
| 32 | 25, 31 | sseqtrri 4032 | 1
⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) |