Step | Hyp | Ref
| Expression |
1 | | pssss 3702 |
. . . . 5
⊢ (𝐹 ⊊ 𝐺 → 𝐹 ⊆ 𝐺) |
2 | | dmss 5323 |
. . . . 5
⊢ (𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝐹 ⊊ 𝐺 → dom 𝐹 ⊆ dom 𝐺) |
4 | 3 | a1i 11 |
. . 3
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊆ dom 𝐺)) |
5 | | pssdif 3945 |
. . . . . . . 8
⊢ (𝐹 ⊊ 𝐺 → (𝐺 ∖ 𝐹) ≠ ∅) |
6 | | n0 3931 |
. . . . . . . 8
⊢ ((𝐺 ∖ 𝐹) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
7 | 5, 6 | sylib 208 |
. . . . . . 7
⊢ (𝐹 ⊊ 𝐺 → ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
8 | 7 | adantl 482 |
. . . . . 6
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
9 | | funrel 5905 |
. . . . . . . . . . 11
⊢ (Fun
𝐺 → Rel 𝐺) |
10 | | reldif 5238 |
. . . . . . . . . . 11
⊢ (Rel
𝐺 → Rel (𝐺 ∖ 𝐹)) |
11 | 9, 10 | syl 17 |
. . . . . . . . . 10
⊢ (Fun
𝐺 → Rel (𝐺 ∖ 𝐹)) |
12 | | elrel 5222 |
. . . . . . . . . . . 12
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → ∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉) |
13 | | eleq1 2689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑝 ∈ (𝐺 ∖ 𝐹) ↔ 〈𝑥, 𝑦〉 ∈ (𝐺 ∖ 𝐹))) |
14 | | df-br 4654 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝐺 ∖ 𝐹)) |
15 | 13, 14 | syl6bbr 278 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑝 ∈ (𝐺 ∖ 𝐹) ↔ 𝑥(𝐺 ∖ 𝐹)𝑦)) |
16 | 15 | biimpcd 239 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ (𝐺 ∖ 𝐹) → (𝑝 = 〈𝑥, 𝑦〉 → 𝑥(𝐺 ∖ 𝐹)𝑦)) |
17 | 16 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → (𝑝 = 〈𝑥, 𝑦〉 → 𝑥(𝐺 ∖ 𝐹)𝑦)) |
18 | 17 | 2eximdv 1848 |
. . . . . . . . . . . 12
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → (∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦)) |
19 | 12, 18 | mpd 15 |
. . . . . . . . . . 11
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦) |
20 | 19 | ex 450 |
. . . . . . . . . 10
⊢ (Rel
(𝐺 ∖ 𝐹) → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦)) |
21 | 11, 20 | syl 17 |
. . . . . . . . 9
⊢ (Fun
𝐺 → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦)) |
22 | 21 | adantr 481 |
. . . . . . . 8
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦)) |
23 | | difss 3737 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∖ 𝐹) ⊆ 𝐺 |
24 | 23 | ssbri 4697 |
. . . . . . . . . . . 12
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → 𝑥𝐺𝑦) |
25 | 24 | eximi 1762 |
. . . . . . . . . . 11
⊢
(∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ∃𝑦 𝑥𝐺𝑦) |
26 | 25 | a1i 11 |
. . . . . . . . . 10
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ∃𝑦 𝑥𝐺𝑦)) |
27 | | brdif 4705 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 ↔ (𝑥𝐺𝑦 ∧ ¬ 𝑥𝐹𝑦)) |
28 | 27 | simprbi 480 |
. . . . . . . . . . . . . 14
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ 𝑥𝐹𝑦) |
29 | 28 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → ¬ 𝑥𝐹𝑦) |
30 | 1 | ssbrd 4696 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ⊊ 𝐺 → (𝑥𝐹𝑧 → 𝑥𝐺𝑧)) |
31 | 30 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → 𝑥𝐺𝑧)) |
32 | 27 | simplbi 476 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → 𝑥𝐺𝑦) |
33 | | dffun2 5898 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (Fun
𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧))) |
34 | 33 | simprbi 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (Fun
𝐺 → ∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
35 | | 2sp 2056 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧) → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
36 | 35 | sps 2055 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧) → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
37 | 34, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
𝐺 → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
38 | | breq2 4657 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧)) |
39 | 38 | biimprd 238 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
40 | 37, 39 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Fun
𝐺 → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
41 | 40 | expd 452 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Fun
𝐺 → (𝑥𝐺𝑦 → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)))) |
42 | 32, 41 | syl5 34 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
𝐺 → (𝑥(𝐺 ∖ 𝐹)𝑦 → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)))) |
43 | 42 | imp 445 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝐺 ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
44 | 43 | adantlr 751 |
. . . . . . . . . . . . . . . 16
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
45 | 44 | com23 86 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → (𝑥𝐺𝑧 → 𝑥𝐹𝑦))) |
46 | 31, 45 | mpdd 43 |
. . . . . . . . . . . . . 14
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
47 | 46 | exlimdv 1861 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (∃𝑧 𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
48 | 29, 47 | mtod 189 |
. . . . . . . . . . . 12
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → ¬ ∃𝑧 𝑥𝐹𝑧) |
49 | 48 | ex 450 |
. . . . . . . . . . 11
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ ∃𝑧 𝑥𝐹𝑧)) |
50 | 49 | exlimdv 1861 |
. . . . . . . . . 10
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ ∃𝑧 𝑥𝐹𝑧)) |
51 | 26, 50 | jcad 555 |
. . . . . . . . 9
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → (∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
52 | 51 | eximdv 1846 |
. . . . . . . 8
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
53 | 22, 52 | syld 47 |
. . . . . . 7
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
54 | 53 | exlimdv 1861 |
. . . . . 6
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
55 | 8, 54 | mpd 15 |
. . . . 5
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
56 | | nss 3663 |
. . . . . 6
⊢ (¬
dom 𝐺 ⊆ dom 𝐹 ↔ ∃𝑥(𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹)) |
57 | | vex 3203 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
58 | 57 | eldm 5321 |
. . . . . . . 8
⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦 𝑥𝐺𝑦) |
59 | 57 | eldm 5321 |
. . . . . . . . 9
⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧) |
60 | 59 | notbii 310 |
. . . . . . . 8
⊢ (¬
𝑥 ∈ dom 𝐹 ↔ ¬ ∃𝑧 𝑥𝐹𝑧) |
61 | 58, 60 | anbi12i 733 |
. . . . . . 7
⊢ ((𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹) ↔ (∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
62 | 61 | exbii 1774 |
. . . . . 6
⊢
(∃𝑥(𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹) ↔ ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
63 | 56, 62 | bitri 264 |
. . . . 5
⊢ (¬
dom 𝐺 ⊆ dom 𝐹 ↔ ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
64 | 55, 63 | sylibr 224 |
. . . 4
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ¬ dom 𝐺 ⊆ dom 𝐹) |
65 | 64 | ex 450 |
. . 3
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → ¬ dom 𝐺 ⊆ dom 𝐹)) |
66 | 4, 65 | jcad 555 |
. 2
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → (dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹))) |
67 | | dfpss3 3693 |
. 2
⊢ (dom
𝐹 ⊊ dom 𝐺 ↔ (dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹)) |
68 | 66, 67 | syl6ibr 242 |
1
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺)) |