Step | Hyp | Ref
| Expression |
1 | | pssss 4024 |
. . . . 5
⊢ (𝐹 ⊊ 𝐺 → 𝐹 ⊆ 𝐺) |
2 | | dmss 5785 |
. . . . 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 4295 |
. . . . . . . 8
⊢ (𝐹 ⊊ 𝐺 → (𝐺 ∖ 𝐹) ≠ ∅) |
6 | | n0 4275 |
. . . . . . . 8
⊢ ((𝐺 ∖ 𝐹) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
7 | 5, 6 | sylib 221 |
. . . . . . 7
⊢ (𝐹 ⊊ 𝐺 → ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
8 | 7 | adantl 485 |
. . . . . 6
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹)) |
9 | | funrel 6414 |
. . . . . . . . . . 11
⊢ (Fun
𝐺 → Rel 𝐺) |
10 | | reldif 5699 |
. . . . . . . . . . 11
⊢ (Rel
𝐺 → Rel (𝐺 ∖ 𝐹)) |
11 | 9, 10 | syl 17 |
. . . . . . . . . 10
⊢ (Fun
𝐺 → Rel (𝐺 ∖ 𝐹)) |
12 | | elrel 5682 |
. . . . . . . . . . . 12
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → ∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉) |
13 | | eleq1 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑝 ∈ (𝐺 ∖ 𝐹) ↔ 〈𝑥, 𝑦〉 ∈ (𝐺 ∖ 𝐹))) |
14 | | df-br 5068 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝐺 ∖ 𝐹)) |
15 | 13, 14 | bitr4di 292 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑝 ∈ (𝐺 ∖ 𝐹) ↔ 𝑥(𝐺 ∖ 𝐹)𝑦)) |
16 | 15 | biimpcd 252 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ (𝐺 ∖ 𝐹) → (𝑝 = 〈𝑥, 𝑦〉 → 𝑥(𝐺 ∖ 𝐹)𝑦)) |
17 | 16 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝐺 ∖ 𝐹) ∧ 𝑝 ∈ (𝐺 ∖ 𝐹)) → (𝑝 = 〈𝑥, 𝑦〉 → 𝑥(𝐺 ∖ 𝐹)𝑦)) |
18 | 17 | 2eximdv 1927 |
. . . . . . . . . . . 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 416 |
. . . . . . . . . 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 484 |
. . . . . . . 8
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦)) |
23 | | difss 4060 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∖ 𝐹) ⊆ 𝐺 |
24 | 23 | ssbri 5112 |
. . . . . . . . . . . 12
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → 𝑥𝐺𝑦) |
25 | 24 | eximi 1842 |
. . . . . . . . . . 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 5120 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 ↔ (𝑥𝐺𝑦 ∧ ¬ 𝑥𝐹𝑦)) |
28 | 27 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ 𝑥𝐹𝑦) |
29 | 28 | adantl 485 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → ¬ 𝑥𝐹𝑦) |
30 | 1 | ssbrd 5110 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 ⊊ 𝐺 → (𝑥𝐹𝑧 → 𝑥𝐺𝑧)) |
31 | 30 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → 𝑥𝐺𝑧)) |
32 | | dffun2 6407 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (Fun
𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧))) |
33 | 32 | simprbi 500 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
𝐺 → ∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
34 | | 2sp 2184 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧) → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
35 | 34 | sps 2183 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑥∀𝑦∀𝑧((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧) → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
36 | 33, 35 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Fun
𝐺 → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → 𝑦 = 𝑧)) |
37 | | breq2 5071 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧)) |
38 | 37 | biimprd 251 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
39 | 36, 38 | syl6 35 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Fun
𝐺 → ((𝑥𝐺𝑦 ∧ 𝑥𝐺𝑧) → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
40 | 39 | expd 419 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
𝐺 → (𝑥𝐺𝑦 → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)))) |
41 | 27 | simplbi 501 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥(𝐺 ∖ 𝐹)𝑦 → 𝑥𝐺𝑦) |
42 | 40, 41 | impel 509 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝐺 ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
43 | 42 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐺𝑧 → (𝑥𝐹𝑧 → 𝑥𝐹𝑦))) |
44 | 43 | com23 86 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → (𝑥𝐺𝑧 → 𝑥𝐹𝑦))) |
45 | 31, 44 | mpdd 43 |
. . . . . . . . . . . . . 14
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
46 | 45 | exlimdv 1941 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → (∃𝑧 𝑥𝐹𝑧 → 𝑥𝐹𝑦)) |
47 | 29, 46 | mtod 201 |
. . . . . . . . . . . 12
⊢ (((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) ∧ 𝑥(𝐺 ∖ 𝐹)𝑦) → ¬ ∃𝑧 𝑥𝐹𝑧) |
48 | 47 | ex 416 |
. . . . . . . . . . 11
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ ∃𝑧 𝑥𝐹𝑧)) |
49 | 48 | exlimdv 1941 |
. . . . . . . . . 10
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ¬ ∃𝑧 𝑥𝐹𝑧)) |
50 | 26, 49 | jcad 516 |
. . . . . . . . 9
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → (∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
51 | 50 | eximdv 1925 |
. . . . . . . 8
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑥∃𝑦 𝑥(𝐺 ∖ 𝐹)𝑦 → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
52 | 22, 51 | syld 47 |
. . . . . . 7
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
53 | 52 | exlimdv 1941 |
. . . . . 6
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → (∃𝑝 𝑝 ∈ (𝐺 ∖ 𝐹) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧))) |
54 | 8, 53 | mpd 15 |
. . . . 5
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
55 | | nss 3977 |
. . . . . 6
⊢ (¬
dom 𝐺 ⊆ dom 𝐹 ↔ ∃𝑥(𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹)) |
56 | | vex 3424 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
57 | 56 | eldm 5783 |
. . . . . . . 8
⊢ (𝑥 ∈ dom 𝐺 ↔ ∃𝑦 𝑥𝐺𝑦) |
58 | 56 | eldm 5783 |
. . . . . . . . 9
⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑧 𝑥𝐹𝑧) |
59 | 58 | notbii 323 |
. . . . . . . 8
⊢ (¬
𝑥 ∈ dom 𝐹 ↔ ¬ ∃𝑧 𝑥𝐹𝑧) |
60 | 57, 59 | anbi12i 630 |
. . . . . . 7
⊢ ((𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹) ↔ (∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
61 | 60 | exbii 1855 |
. . . . . 6
⊢
(∃𝑥(𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹) ↔ ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
62 | 55, 61 | bitri 278 |
. . . . 5
⊢ (¬
dom 𝐺 ⊆ dom 𝐹 ↔ ∃𝑥(∃𝑦 𝑥𝐺𝑦 ∧ ¬ ∃𝑧 𝑥𝐹𝑧)) |
63 | 54, 62 | sylibr 237 |
. . . 4
⊢ ((Fun
𝐺 ∧ 𝐹 ⊊ 𝐺) → ¬ dom 𝐺 ⊆ dom 𝐹) |
64 | 63 | ex 416 |
. . 3
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → ¬ dom 𝐺 ⊆ dom 𝐹)) |
65 | 4, 64 | jcad 516 |
. 2
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → (dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹))) |
66 | | dfpss3 4015 |
. 2
⊢ (dom
𝐹 ⊊ dom 𝐺 ↔ (dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹)) |
67 | 65, 66 | syl6ibr 255 |
1
⊢ (Fun
𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺)) |