Proof of Theorem fiinfi
Step | Hyp | Ref
| Expression |
1 | | fiinfi.a |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) |
2 | | elinel1 4130 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴) |
3 | | elinel1 4130 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) → 𝑦 ∈ 𝐴) |
4 | 3 | imim1i 63 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐴)) |
5 | 4 | ralimi2 3082 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴) |
6 | 2, 5 | imim12i 62 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴 ∩ 𝐵) → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴)) |
7 | 6 | ralimi2 3082 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴) |
8 | 1, 7 | syl 17 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴) |
9 | | fiinfi.b |
. . . . . . 7
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) |
10 | | elinel2 4131 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵) |
11 | | elinel2 4131 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) → 𝑦 ∈ 𝐵) |
12 | 11 | imim1i 63 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐵 → (𝑥 ∩ 𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵)) |
13 | 12 | ralimi2 3082 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵) |
14 | 10, 13 | imim12i 62 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵)) |
15 | 14 | ralimi2 3082 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵) |
16 | 9, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵) |
17 | | r19.26-2 3094 |
. . . . . 6
⊢
(∀𝑥 ∈
(𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵)) |
18 | 8, 16, 17 | sylanbrc 582 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵)) |
19 | | elin 3904 |
. . . . . 6
⊢ ((𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵) ↔ ((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵)) |
20 | 19 | 2ralbii 3090 |
. . . . 5
⊢
(∀𝑥 ∈
(𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵)) |
21 | 18, 20 | sylibr 233 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵)) |
22 | | fiinfi.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 = (𝐴 ∩ 𝐵)) |
23 | 22 | eleq2d 2822 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∩ 𝑦) ∈ 𝐶 ↔ (𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵))) |
24 | 23 | ralbidv 3119 |
. . . . 5
⊢ (𝜑 → (∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵))) |
25 | 24 | ralbidv 3119 |
. . . 4
⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵))) |
26 | 21, 25 | mpbird 256 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶) |
27 | 22 | raleqdv 3343 |
. . . 4
⊢ (𝜑 → (∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶)) |
28 | 27 | ralbidv 3119 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶)) |
29 | 26, 28 | mpbird 256 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶) |
30 | 22 | raleqdv 3343 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶)) |
31 | 29, 30 | mpbird 256 |
1
⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶) |