Step | Hyp | Ref
| Expression |
1 | | frgrhash2wsp.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | fusgreg2wsp.m |
. . . . . 6
⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) |
3 | 1, 2 | fusgreg2wsplem 27741 |
. . . . 5
⊢ (𝑁 ∈ 𝑉 → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁))) |
4 | 3 | adantl 475 |
. . . 4
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁))) |
5 | 1 | wspthsnwspthsnon 27296 |
. . . . . . 7
⊢ (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦)) |
6 | | fusgrusgr 26669 |
. . . . . . . . . 10
⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈
USGraph) |
7 | 6 | adantr 474 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph) |
8 | | eqid 2778 |
. . . . . . . . . 10
⊢
(Edg‘𝐺) =
(Edg‘𝐺) |
9 | 1, 8 | usgr2wspthon 27345 |
. . . . . . . . 9
⊢ ((𝐺 ∈ USGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
10 | 7, 9 | sylan 575 |
. . . . . . . 8
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
11 | 10 | 2rexbidva 3241 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
12 | 5, 11 | syl5bb 275 |
. . . . . 6
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
13 | 12 | anbi1d 623 |
. . . . 5
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁))) |
14 | | 19.41vv 1993 |
. . . . . . 7
⊢
(∃𝑥∃𝑦(((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁)) |
15 | | velsn 4414 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ {〈“𝑥𝑁𝑦”〉} ↔ 𝑧 = 〈“𝑥𝑁𝑦”〉) |
16 | 15 | bicomi 216 |
. . . . . . . . . . 11
⊢ (𝑧 = 〈“𝑥𝑁𝑦”〉 ↔ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}) |
17 | 16 | anbi2i 616 |
. . . . . . . . . 10
⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉})) |
18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}))) |
19 | | simplr 759 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁 ∈ 𝑉) |
20 | | anass 462 |
. . . . . . . . . . . . . . 15
⊢ (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ (𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
21 | | ancom 454 |
. . . . . . . . . . . . . . 15
⊢ ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ (𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = 〈“𝑥𝑚𝑦”〉)) |
22 | | an12 635 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) |
23 | | nesym 3025 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥) |
24 | | prcom 4499 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑚, 𝑦} = {𝑦, 𝑚} |
25 | 24 | eleq1i 2850 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺)) |
26 | 23, 25 | anbi12ci 621 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) |
27 | 26 | anbi2i 616 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))) |
28 | 22, 27 | bitri 267 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))) |
29 | 28 | anbi1i 617 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = 〈“𝑥𝑚𝑦”〉) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑚𝑦”〉)) |
30 | 20, 21, 29 | 3bitri 289 |
. . . . . . . . . . . . . 14
⊢ (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑚𝑦”〉)) |
31 | | preq2 4501 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁}) |
32 | 31 | eleq1d 2844 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺))) |
33 | | preq2 4501 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁}) |
34 | 33 | eleq1d 2844 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺))) |
35 | 34 | anbi1d 623 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))) |
36 | 32, 35 | anbi12d 624 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑁 → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))) |
37 | | s3eq2 14021 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑁 → 〈“𝑥𝑚𝑦”〉 = 〈“𝑥𝑁𝑦”〉) |
38 | 37 | eqeq2d 2788 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑁 → (𝑧 = 〈“𝑥𝑚𝑦”〉 ↔ 𝑧 = 〈“𝑥𝑁𝑦”〉)) |
39 | 36, 38 | anbi12d 624 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑁 → ((({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑚𝑦”〉) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉))) |
40 | 30, 39 | syl5bb 275 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑁 → (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉))) |
41 | 40 | adantl 475 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ 𝑚 = 𝑁) → (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉))) |
42 | | fveq1 6445 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 〈“𝑥𝑚𝑦”〉 → (𝑧‘1) = (〈“𝑥𝑚𝑦”〉‘1)) |
43 | | vex 3401 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑚 ∈ V |
44 | | s3fv1 14043 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ V →
(〈“𝑥𝑚𝑦”〉‘1) = 𝑚) |
45 | 43, 44 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈“𝑥𝑚𝑦”〉‘1) = 𝑚 |
46 | 42, 45 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 〈“𝑥𝑚𝑦”〉 → (𝑧‘1) = 𝑚) |
47 | 46 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 〈“𝑥𝑚𝑦”〉 → ((𝑧‘1) = 𝑁 ↔ 𝑚 = 𝑁)) |
48 | 47 | biimpd 221 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 〈“𝑥𝑚𝑦”〉 → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁)) |
49 | 48 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁)) |
50 | 49 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁)) |
51 | 50 | com12 32 |
. . . . . . . . . . . . . 14
⊢ ((𝑧‘1) = 𝑁 → (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁)) |
52 | 51 | ad2antll 719 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → (((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁)) |
53 | 52 | imp 397 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → 𝑚 = 𝑁) |
54 | 19, 41, 53 | rspcebdv 3516 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → (∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉))) |
55 | 54 | pm5.32da 574 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉)))) |
56 | | an32 636 |
. . . . . . . . . . 11
⊢ ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
57 | 56 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))) |
58 | | usgrumgr 26528 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 ∈ USGraph → 𝐺 ∈
UMGraph) |
59 | 1, 8 | umgrpredgv 26489 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
60 | 59 | simpld 490 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥 ∈ 𝑉) |
61 | 60 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥 ∈ 𝑉)) |
62 | 1, 8 | umgrpredgv 26489 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
63 | 62 | simpld 490 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦 ∈ 𝑉) |
64 | 63 | expcom 404 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉)) |
65 | 64 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉)) |
66 | 65 | com12 32 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ 𝑉)) |
67 | 61, 66 | anim12d 602 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 ∈ UMGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
68 | 6, 58, 67 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 ∈ FinUSGraph →
(({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
69 | 68 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
70 | 69 | com12 32 |
. . . . . . . . . . . . . . 15
⊢ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
71 | 70 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) |
72 | 71 | impcom 398 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) |
73 | | fveq1 6445 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 〈“𝑥𝑁𝑦”〉 → (𝑧‘1) = (〈“𝑥𝑁𝑦”〉‘1)) |
74 | 73 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) → (𝑧‘1) = (〈“𝑥𝑁𝑦”〉‘1)) |
75 | | s3fv1 14043 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ 𝑉 → (〈“𝑥𝑁𝑦”〉‘1) = 𝑁) |
76 | 75 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (〈“𝑥𝑁𝑦”〉‘1) = 𝑁) |
77 | 74, 76 | sylan9eqr 2836 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉)) → (𝑧‘1) = 𝑁) |
78 | 72, 77 | jca 507 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) |
79 | 78 | ex 403 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁))) |
80 | 79 | pm4.71rd 558 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉)))) |
81 | 55, 57, 80 | 3bitr4d 303 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = 〈“𝑥𝑁𝑦”〉))) |
82 | 8 | nbusgreledg 26700 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺))) |
83 | 6, 82 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺))) |
84 | 83 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺))) |
85 | | eldif 3802 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥})) |
86 | 8 | nbusgreledg 26700 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺))) |
87 | 6, 86 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺))) |
88 | 87 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺))) |
89 | | velsn 4414 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) |
90 | 89 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)) |
91 | 90 | notbid 310 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥)) |
92 | 88, 91 | anbi12d 624 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))) |
93 | 85, 92 | syl5bb 275 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))) |
94 | 84, 93 | anbi12d 624 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))) |
95 | 94 | anbi1d 623 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}))) |
96 | 18, 81, 95 | 3bitr4d 303 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}))) |
97 | 96 | 2exbidv 1967 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥∃𝑦(((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}))) |
98 | 14, 97 | syl5bbr 277 |
. . . . . 6
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉}))) |
99 | | r2ex 3246 |
. . . . . . 7
⊢
(∃𝑥 ∈
𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) |
100 | 99 | anbi1i 617 |
. . . . . 6
⊢
((∃𝑥 ∈
𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁)) |
101 | | r2ex 3246 |
. . . . . 6
⊢
(∃𝑥 ∈
(𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉} ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉})) |
102 | 98, 100, 101 | 3bitr4g 306 |
. . . . 5
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = 〈“𝑥𝑚𝑦”〉 ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉})) |
103 | | vex 3401 |
. . . . . . . 8
⊢ 𝑧 ∈ V |
104 | | eleq1w 2842 |
. . . . . . . . 9
⊢ (𝑝 = 𝑧 → (𝑝 ∈ {〈“𝑥𝑁𝑦”〉} ↔ 𝑧 ∈ {〈“𝑥𝑁𝑦”〉})) |
105 | 104 | 2rexbidv 3242 |
. . . . . . . 8
⊢ (𝑝 = 𝑧 → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉})) |
106 | 103, 105 | elab 3558 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉}) |
107 | 106 | bicomi 216 |
. . . . . 6
⊢
(∃𝑥 ∈
(𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}}) |
108 | 107 | a1i 11 |
. . . . 5
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {〈“𝑥𝑁𝑦”〉} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}})) |
109 | 13, 102, 108 | 3bitrd 297 |
. . . 4
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}})) |
110 | 4, 109 | bitrd 271 |
. . 3
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}})) |
111 | 110 | eqrdv 2776 |
. 2
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}}) |
112 | | dfiunv2 4789 |
. 2
⊢ ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){〈“𝑥𝑁𝑦”〉} = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {〈“𝑥𝑁𝑦”〉}} |
113 | 111, 112 | syl6eqr 2832 |
1
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪
𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){〈“𝑥𝑁𝑦”〉}) |