Step | Hyp | Ref
| Expression |
1 | | imainlem 5244 |
. . 3
⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) |
2 | 1 | a1i 9 |
. 2
⊢ (Fun
◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))) |
3 | | eeanv 1909 |
. . . . . 6
⊢
(∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) |
4 | | simprll 527 |
. . . . . . . . . . 11
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐴) |
5 | | simpr 109 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥𝐹𝑦) |
6 | | simpr 109 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦) → 𝑧𝐹𝑦) |
7 | 5, 6 | anim12i 336 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦)) |
8 | | funcnveq 5226 |
. . . . . . . . . . . . . . . . 17
⊢ (Fun
◡𝐹 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧)) |
9 | 8 | biimpi 119 |
. . . . . . . . . . . . . . . 16
⊢ (Fun
◡𝐹 → ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧)) |
10 | 9 | 19.21bi 1535 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡𝐹 → ∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧)) |
11 | 10 | 19.21bbi 1536 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡𝐹 → ((𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦) → 𝑥 = 𝑧)) |
12 | 11 | imp 123 |
. . . . . . . . . . . . 13
⊢ ((Fun
◡𝐹 ∧ (𝑥𝐹𝑦 ∧ 𝑧𝐹𝑦)) → 𝑥 = 𝑧) |
13 | 7, 12 | sylan2 284 |
. . . . . . . . . . . 12
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 = 𝑧) |
14 | | simprrl 529 |
. . . . . . . . . . . 12
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑧 ∈ 𝐵) |
15 | 13, 14 | eqeltrd 2231 |
. . . . . . . . . . 11
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ 𝐵) |
16 | | elin 3286 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
17 | 4, 15, 16 | sylanbrc 414 |
. . . . . . . . . 10
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴 ∩ 𝐵)) |
18 | | simprlr 528 |
. . . . . . . . . 10
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → 𝑥𝐹𝑦) |
19 | 17, 18 | jca 304 |
. . . . . . . . 9
⊢ ((Fun
◡𝐹 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)) |
20 | 19 | ex 114 |
. . . . . . . 8
⊢ (Fun
◡𝐹 → (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))) |
21 | 20 | exlimdv 1796 |
. . . . . . 7
⊢ (Fun
◡𝐹 → (∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))) |
22 | 21 | eximdv 1857 |
. . . . . 6
⊢ (Fun
◡𝐹 → (∃𝑥∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))) |
23 | 3, 22 | syl5bir 152 |
. . . . 5
⊢ (Fun
◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦))) |
24 | | df-rex 2438 |
. . . . . 6
⊢
(∃𝑥 ∈
𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)) |
25 | | df-rex 2438 |
. . . . . 6
⊢
(∃𝑧 ∈
𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦)) |
26 | 24, 25 | anbi12i 456 |
. . . . 5
⊢
((∃𝑥 ∈
𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑧𝐹𝑦))) |
27 | | df-rex 2438 |
. . . . 5
⊢
(∃𝑥 ∈
(𝐴 ∩ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥𝐹𝑦)) |
28 | 23, 26, 27 | 3imtr4g 204 |
. . . 4
⊢ (Fun
◡𝐹 → ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦)) |
29 | 28 | ss2abdv 3197 |
. . 3
⊢ (Fun
◡𝐹 → {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦}) |
30 | | dfima2 4923 |
. . . . 5
⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} |
31 | | dfima2 4923 |
. . . . 5
⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦} |
32 | 30, 31 | ineq12i 3302 |
. . . 4
⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}) |
33 | | inab 3371 |
. . . 4
⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} |
34 | 32, 33 | eqtri 2175 |
. . 3
⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ∃𝑧 ∈ 𝐵 𝑧𝐹𝑦)} |
35 | | dfima2 4923 |
. . 3
⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝐹𝑦} |
36 | 29, 34, 35 | 3sstr4g 3167 |
. 2
⊢ (Fun
◡𝐹 → ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∩ 𝐵))) |
37 | 2, 36 | eqssd 3141 |
1
⊢ (Fun
◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))) |