Step | Hyp | Ref
| Expression |
1 | | rexcom4 3172 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑦∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
2 | | rexcom4 3172 |
. . . . . . . . . 10
⊢
(∃𝑖 ∈
𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ ∃𝑦∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) |
3 | 1, 2 | orbi12i 915 |
. . . . . . . . 9
⊢
((∃𝑣 ∈
𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (∃𝑦∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑦∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
4 | | 19.43 1890 |
. . . . . . . . 9
⊢
(∃𝑦(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (∃𝑦∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑦∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
5 | 3, 4 | bitr4i 281 |
. . . . . . . 8
⊢
((∃𝑣 ∈
𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑦(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
6 | 5 | rexbii 3170 |
. . . . . . 7
⊢
(∃𝑢 ∈
(𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑢 ∈ (𝑈 ∖ 𝑆)∃𝑦(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
7 | | rexcom4 3172 |
. . . . . . 7
⊢
(∃𝑢 ∈
(𝑈 ∖ 𝑆)∃𝑦(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑦∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
8 | 6, 7 | bitri 278 |
. . . . . 6
⊢
(∃𝑢 ∈
(𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑦∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
9 | | rexcom4 3172 |
. . . . . . . 8
⊢
(∃𝑣 ∈
(𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑦∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
10 | 9 | rexbii 3170 |
. . . . . . 7
⊢
(∃𝑢 ∈
𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑢 ∈ 𝑆 ∃𝑦∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
11 | | rexcom4 3172 |
. . . . . . 7
⊢
(∃𝑢 ∈
𝑆 ∃𝑦∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑦∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
12 | 10, 11 | bitri 278 |
. . . . . 6
⊢
(∃𝑢 ∈
𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑦∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
13 | 8, 12 | orbi12i 915 |
. . . . 5
⊢
((∃𝑢 ∈
(𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑦∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
14 | | 19.43 1890 |
. . . . 5
⊢
(∃𝑦(∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑦∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
15 | 13, 14 | bitr4i 281 |
. . . 4
⊢
((∃𝑢 ∈
(𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
16 | | difssd 4047 |
. . . . . . . 8
⊢ (𝑆 ⊆ 𝑈 → (𝑈 ∖ 𝑆) ⊆ 𝑈) |
17 | | ssralv 3967 |
. . . . . . . 8
⊢ ((𝑈 ∖ 𝑆) ⊆ 𝑈 → (∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑢 ∈ (𝑈 ∖ 𝑆)(∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊))) |
18 | 16, 17 | syl 17 |
. . . . . . 7
⊢ (𝑆 ⊆ 𝑈 → (∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑢 ∈ (𝑈 ∖ 𝑆)(∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊))) |
19 | 18 | impcom 411 |
. . . . . 6
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → ∀𝑢 ∈ (𝑈 ∖ 𝑆)(∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊)) |
20 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑧 = 𝐴) |
21 | 20 | exlimiv 1938 |
. . . . . . . . . . 11
⊢
(∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑧 = 𝐴) |
22 | | elisset 2819 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ 𝑋 → ∃𝑦 𝑦 = 𝐵) |
23 | | ibar 532 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝐴 → (𝑦 = 𝐵 ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
24 | 23 | bicomd 226 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝐴 → ((𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ 𝑦 = 𝐵)) |
25 | 24 | exbidv 1929 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝐴 → (∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑦 𝑦 = 𝐵)) |
26 | 22, 25 | syl5ibrcom 250 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ 𝑋 → (𝑧 = 𝐴 → ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
27 | 21, 26 | impbid2 229 |
. . . . . . . . . 10
⊢ (𝐵 ∈ 𝑋 → (∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ 𝑧 = 𝐴)) |
28 | 27 | ralrexbid 3241 |
. . . . . . . . 9
⊢
(∀𝑣 ∈
𝑈 𝐵 ∈ 𝑋 → (∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑣 ∈ 𝑈 𝑧 = 𝐴)) |
29 | 28 | adantr 484 |
. . . . . . . 8
⊢
((∀𝑣 ∈
𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → (∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑣 ∈ 𝑈 𝑧 = 𝐴)) |
30 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑧 = 𝐶 ∧ 𝑦 = 𝐷) → 𝑧 = 𝐶) |
31 | 30 | exlimiv 1938 |
. . . . . . . . . . 11
⊢
(∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) → 𝑧 = 𝐶) |
32 | | elisset 2819 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐷) |
33 | | ibar 532 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝐶 → (𝑦 = 𝐷 ↔ (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
34 | 33 | bicomd 226 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝐶 → ((𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ 𝑦 = 𝐷)) |
35 | 34 | exbidv 1929 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝐶 → (∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ ∃𝑦 𝑦 = 𝐷)) |
36 | 32, 35 | syl5ibrcom 250 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑊 → (𝑧 = 𝐶 → ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
37 | 31, 36 | impbid2 229 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑊 → (∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ 𝑧 = 𝐶)) |
38 | 37 | ralrexbid 3241 |
. . . . . . . . 9
⊢
(∀𝑖 ∈
𝐼 𝐷 ∈ 𝑊 → (∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶)) |
39 | 38 | adantl 485 |
. . . . . . . 8
⊢
((∀𝑣 ∈
𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → (∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷) ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶)) |
40 | 29, 39 | orbi12d 919 |
. . . . . . 7
⊢
((∀𝑣 ∈
𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ((∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶))) |
41 | 40 | ralrexbid 3241 |
. . . . . 6
⊢
(∀𝑢 ∈
(𝑈 ∖ 𝑆)(∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶))) |
42 | 19, 41 | syl 17 |
. . . . 5
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶))) |
43 | | ssralv 3967 |
. . . . . . . 8
⊢ (𝑆 ⊆ 𝑈 → (∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑢 ∈ 𝑆 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊))) |
44 | | ssralv 3967 |
. . . . . . . . . . 11
⊢ ((𝑈 ∖ 𝑆) ⊆ 𝑈 → (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 → ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋)) |
45 | 16, 44 | syl 17 |
. . . . . . . . . 10
⊢ (𝑆 ⊆ 𝑈 → (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 → ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋)) |
46 | 45 | adantrd 495 |
. . . . . . . . 9
⊢ (𝑆 ⊆ 𝑈 → ((∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋)) |
47 | 46 | ralimdv 3101 |
. . . . . . . 8
⊢ (𝑆 ⊆ 𝑈 → (∀𝑢 ∈ 𝑆 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋)) |
48 | 43, 47 | syld 47 |
. . . . . . 7
⊢ (𝑆 ⊆ 𝑈 → (∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋)) |
49 | 48 | impcom 411 |
. . . . . 6
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋) |
50 | 27 | ralrexbid 3241 |
. . . . . . 7
⊢
(∀𝑣 ∈
(𝑈 ∖ 𝑆)𝐵 ∈ 𝑋 → (∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴)) |
51 | 50 | ralrexbid 3241 |
. . . . . 6
⊢
(∀𝑢 ∈
𝑆 ∀𝑣 ∈ (𝑈 ∖ 𝑆)𝐵 ∈ 𝑋 → (∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴)) |
52 | 49, 51 | syl 17 |
. . . . 5
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → (∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴)) |
53 | 42, 52 | orbi12d 919 |
. . . 4
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → ((∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 ∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 ∃𝑦(𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)∃𝑦(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴))) |
54 | 15, 53 | bitr3id 288 |
. . 3
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → (∃𝑦(∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴))) |
55 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) |
56 | 55 | anbi1d 633 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
57 | 56 | rexbidv 3216 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
58 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶)) |
59 | 58 | anbi1d 633 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
60 | 59 | rexbidv 3216 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷))) |
61 | 57, 60 | orbi12d 919 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → ((∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)))) |
62 | 61 | rexbidv 3216 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ ∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)))) |
63 | 56 | 2rexbidv 3219 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
64 | 62, 63 | orbi12d 919 |
. . . . 5
⊢ (𝑥 = 𝑧 → ((∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))) |
65 | 64 | dmopabelb 5785 |
. . . 4
⊢ (𝑧 ∈ V → (𝑧 ∈ dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} ↔ ∃𝑦(∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))) |
66 | 65 | elv 3414 |
. . 3
⊢ (𝑧 ∈ dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} ↔ ∃𝑦(∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑧 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
67 | | vex 3412 |
. . . 4
⊢ 𝑧 ∈ V |
68 | 55 | rexbidv 3216 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ↔ ∃𝑣 ∈ 𝑈 𝑧 = 𝐴)) |
69 | 58 | rexbidv 3216 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (∃𝑖 ∈ 𝐼 𝑥 = 𝐶 ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶)) |
70 | 68, 69 | orbi12d 919 |
. . . . . 6
⊢ (𝑥 = 𝑧 → ((∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ↔ (∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶))) |
71 | 70 | rexbidv 3216 |
. . . . 5
⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ↔ ∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶))) |
72 | 55 | 2rexbidv 3219 |
. . . . 5
⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴 ↔ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴)) |
73 | 71, 72 | orbi12d 919 |
. . . 4
⊢ (𝑥 = 𝑧 → ((∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴) ↔ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴))) |
74 | 67, 73 | elab 3587 |
. . 3
⊢ (𝑧 ∈ {𝑥 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴)} ↔ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑧 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑧 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑧 = 𝐴)) |
75 | 54, 66, 74 | 3bitr4g 317 |
. 2
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → (𝑧 ∈ dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} ↔ 𝑧 ∈ {𝑥 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴)})) |
76 | 75 | eqrdv 2735 |
1
⊢
((∀𝑢 ∈
𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴)}) |