Step | Hyp | Ref
| Expression |
1 | | rngop.1 |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
2 | 1 | rnmpt2 5611 |
. . . 4
⊢ ran 𝐹 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶} |
3 | 2 | raleqi 2509 |
. . 3
⊢
(∀𝑧 ∈
ran 𝐹𝜑 ↔ ∀𝑧 ∈ {𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑) |
4 | | eqeq1 2046 |
. . . . 5
⊢ (𝑤 = 𝑧 → (𝑤 = 𝐶 ↔ 𝑧 = 𝐶)) |
5 | 4 | 2rexbidv 2349 |
. . . 4
⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)) |
6 | 5 | ralab 2701 |
. . 3
⊢
(∀𝑧 ∈
{𝑤 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑤 = 𝐶}𝜑 ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
7 | | ralcom4 2576 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
8 | | r19.23v 2425 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
9 | 8 | albii 1359 |
. . . 4
⊢
(∀𝑧∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
10 | 7, 9 | bitr2i 174 |
. . 3
⊢
(∀𝑧(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
11 | 3, 6, 10 | 3bitri 195 |
. 2
⊢
(∀𝑧 ∈
ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
12 | | ralcom4 2576 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑)) |
13 | | r19.23v 2425 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝐵 (𝑧 = 𝐶 → 𝜑) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
14 | 13 | albii 1359 |
. . . . . 6
⊢
(∀𝑧∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
15 | 12, 14 | bitri 173 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑)) |
16 | | nfv 1421 |
. . . . . . . 8
⊢
Ⅎ𝑧𝜓 |
17 | | ralrnmpt2.2 |
. . . . . . . 8
⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) |
18 | 16, 17 | ceqsalg 2582 |
. . . . . . 7
⊢ (𝐶 ∈ 𝑉 → (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓)) |
19 | 18 | ralimi 2384 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓)) |
20 | | ralbi 2445 |
. . . . . 6
⊢
(∀𝑦 ∈
𝐵 (∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
21 | 19, 20 | syl 14 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
22 | 15, 21 | syl5bbr 183 |
. . . 4
⊢
(∀𝑦 ∈
𝐵 𝐶 ∈ 𝑉 → (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
23 | 22 | ralimi 2384 |
. . 3
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓)) |
24 | | ralbi 2445 |
. . 3
⊢
(∀𝑥 ∈
𝐴 (∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑦 ∈ 𝐵 𝜓) → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) |
25 | 23, 24 | syl 14 |
. 2
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑧(∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) |
26 | 11, 25 | syl5bb 181 |
1
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) |