Proof of Theorem ov2gf
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elex 3501 | . . 3
⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V) | 
| 2 |  | ov2gf.a | . . . 4
⊢
Ⅎ𝑥𝐴 | 
| 3 |  | ov2gf.c | . . . 4
⊢
Ⅎ𝑦𝐴 | 
| 4 |  | ov2gf.d | . . . 4
⊢
Ⅎ𝑦𝐵 | 
| 5 |  | ov2gf.1 | . . . . . 6
⊢
Ⅎ𝑥𝐺 | 
| 6 | 5 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑥 𝐺 ∈ V | 
| 7 |  | ov2gf.5 | . . . . . . . 8
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | 
| 8 |  | nfmpo1 7513 | . . . . . . . 8
⊢
Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | 
| 9 | 7, 8 | nfcxfr 2903 | . . . . . . 7
⊢
Ⅎ𝑥𝐹 | 
| 10 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑥𝑦 | 
| 11 | 2, 9, 10 | nfov 7461 | . . . . . 6
⊢
Ⅎ𝑥(𝐴𝐹𝑦) | 
| 12 | 11, 5 | nfeq 2919 | . . . . 5
⊢
Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺 | 
| 13 | 6, 12 | nfim 1896 | . . . 4
⊢
Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) | 
| 14 |  | ov2gf.2 | . . . . . 6
⊢
Ⅎ𝑦𝑆 | 
| 15 | 14 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑦 𝑆 ∈ V | 
| 16 |  | nfmpo2 7514 | . . . . . . . 8
⊢
Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | 
| 17 | 7, 16 | nfcxfr 2903 | . . . . . . 7
⊢
Ⅎ𝑦𝐹 | 
| 18 | 3, 17, 4 | nfov 7461 | . . . . . 6
⊢
Ⅎ𝑦(𝐴𝐹𝐵) | 
| 19 | 18, 14 | nfeq 2919 | . . . . 5
⊢
Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆 | 
| 20 | 15, 19 | nfim 1896 | . . . 4
⊢
Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆) | 
| 21 |  | ov2gf.3 | . . . . . 6
⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) | 
| 22 | 21 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V)) | 
| 23 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | 
| 24 | 23, 21 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺)) | 
| 25 | 22, 24 | imbi12d 344 | . . . 4
⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺))) | 
| 26 |  | ov2gf.4 | . . . . . 6
⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) | 
| 27 | 26 | eleq1d 2826 | . . . . 5
⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V)) | 
| 28 |  | oveq2 7439 | . . . . . 6
⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | 
| 29 | 28, 26 | eqeq12d 2753 | . . . . 5
⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆)) | 
| 30 | 27, 29 | imbi12d 344 | . . . 4
⊢ (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))) | 
| 31 | 7 | ovmpt4g 7580 | . . . . 5
⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅) | 
| 32 | 31 | 3expia 1122 | . . . 4
⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅)) | 
| 33 | 2, 3, 4, 13, 20, 25, 30, 32 | vtocl2gaf 3579 | . . 3
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)) | 
| 34 | 1, 33 | syl5 34 | . 2
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ 𝐻 → (𝐴𝐹𝐵) = 𝑆)) | 
| 35 | 34 | 3impia 1118 | 1
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |