Step | Hyp | Ref
| Expression |
1 | | anandir 586 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
2 | 1 | exbii 1598 |
. . . . . . 7
⊢
(∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
3 | | 19.40 1624 |
. . . . . . 7
⊢
(∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
4 | 2, 3 | sylbi 120 |
. . . . . 6
⊢
(∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
5 | | nfv 1521 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥Fun ◡𝐹 |
6 | | nfe1 1489 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) |
7 | 5, 6 | nfan 1558 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(Fun ◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) |
8 | | funmo 5211 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡𝐹 → ∃*𝑥 𝑦◡𝐹𝑥) |
9 | | vex 2733 |
. . . . . . . . . . . . . . . 16
⊢ 𝑦 ∈ V |
10 | | vex 2733 |
. . . . . . . . . . . . . . . 16
⊢ 𝑥 ∈ V |
11 | 9, 10 | brcnv 4792 |
. . . . . . . . . . . . . . 15
⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦) |
12 | 11 | mobii 2056 |
. . . . . . . . . . . . . 14
⊢
(∃*𝑥 𝑦◡𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦) |
13 | 8, 12 | sylib 121 |
. . . . . . . . . . . . 13
⊢ (Fun
◡𝐹 → ∃*𝑥 𝑥𝐹𝑦) |
14 | | mopick 2097 |
. . . . . . . . . . . . 13
⊢
((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵)) |
15 | 13, 14 | sylan 281 |
. . . . . . . . . . . 12
⊢ ((Fun
◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵)) |
16 | 15 | con2d 619 |
. . . . . . . . . . 11
⊢ ((Fun
◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦)) |
17 | | imnan 685 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
18 | 16, 17 | sylib 121 |
. . . . . . . . . 10
⊢ ((Fun
◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
19 | 7, 18 | alrimi 1515 |
. . . . . . . . 9
⊢ ((Fun
◡𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵)) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
20 | 19 | ex 114 |
. . . . . . . 8
⊢ (Fun
◡𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) → ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
21 | | exancom 1601 |
. . . . . . . 8
⊢
(∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
22 | | alnex 1492 |
. . . . . . . 8
⊢
(∀𝑥 ¬
(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
23 | 20, 21, 22 | 3imtr3g 203 |
. . . . . . 7
⊢ (Fun
◡𝐹 → (∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
24 | 23 | anim2d 335 |
. . . . . 6
⊢ (Fun
◡𝐹 → ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∃𝑥(¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))) |
25 | 4, 24 | syl5 32 |
. . . . 5
⊢ (Fun
◡𝐹 → (∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))) |
26 | | df-rex 2454 |
. . . . . 6
⊢
(∃𝑥 ∈
(𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦)) |
27 | | eldif 3130 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
28 | 27 | anbi1i 455 |
. . . . . . 7
⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦)) |
29 | 28 | exbii 1598 |
. . . . . 6
⊢
(∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦)) |
30 | 26, 29 | bitri 183 |
. . . . 5
⊢
(∃𝑥 ∈
(𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦)) |
31 | | df-rex 2454 |
. . . . . 6
⊢
(∃𝑥 ∈
𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)) |
32 | | df-rex 2454 |
. . . . . . 7
⊢
(∃𝑥 ∈
𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
33 | 32 | notbii 663 |
. . . . . 6
⊢ (¬
∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) |
34 | 31, 33 | anbi12i 457 |
. . . . 5
⊢
((∃𝑥 ∈
𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))) |
35 | 25, 30, 34 | 3imtr4g 204 |
. . . 4
⊢ (Fun
◡𝐹 → (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 → (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦))) |
36 | 35 | ss2abdv 3220 |
. . 3
⊢ (Fun
◡𝐹 → {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}) |
37 | | dfima2 4953 |
. . 3
⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦} |
38 | | dfima2 4953 |
. . . . 5
⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} |
39 | | dfima2 4953 |
. . . . 5
⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦} |
40 | 38, 39 | difeq12i 3243 |
. . . 4
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) |
41 | | difab 3396 |
. . . 4
⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)} |
42 | 40, 41 | eqtri 2191 |
. . 3
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)} |
43 | 36, 37, 42 | 3sstr4g 3190 |
. 2
⊢ (Fun
◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))) |
44 | | imadiflem 5275 |
. . 3
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)) |
45 | 44 | a1i 9 |
. 2
⊢ (Fun
◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))) |
46 | 43, 45 | eqssd 3164 |
1
⊢ (Fun
◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))) |