Proof of Theorem incistruhgr
Step | Hyp | Ref
| Expression |
1 | | id 22 |
. . . . . . . . . 10
⊢ (𝑉 = 𝑃 → 𝑉 = 𝑃) |
2 | 1 | rabeqdv 3167 |
. . . . . . . . 9
⊢ (𝑉 = 𝑃 → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) |
3 | 2 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑉 = 𝑃 → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) |
4 | 3 | eqeq2d 2620 |
. . . . . . 7
⊢ (𝑉 = 𝑃 → (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) |
5 | | xpeq1 5052 |
. . . . . . . . 9
⊢ (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿)) |
6 | 5 | sseq2d 3596 |
. . . . . . . 8
⊢ (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿))) |
7 | 6 | 3anbi2d 1396 |
. . . . . . 7
⊢ (𝑉 = 𝑃 → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))) |
8 | 4, 7 | anbi12d 743 |
. . . . . 6
⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))) |
9 | | dmeq 5246 |
. . . . . . . . 9
⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})) |
10 | | incistruhgr.v |
. . . . . . . . . . . . 13
⊢ 𝑉 = (Vtx‘𝐺) |
11 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢
(Vtx‘𝐺) ∈
V |
12 | 10, 11 | eqeltri 2684 |
. . . . . . . . . . . 12
⊢ 𝑉 ∈ V |
13 | 12 | rabex 4740 |
. . . . . . . . . . 11
⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ V |
14 | | eqid 2610 |
. . . . . . . . . . 11
⊢ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) |
15 | 13, 14 | dmmpti 5936 |
. . . . . . . . . 10
⊢ dom
(𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = 𝐿 |
16 | 15 | a1i 11 |
. . . . . . . . 9
⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = 𝐿) |
17 | 9, 16 | eqtrd 2644 |
. . . . . . . 8
⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = 𝐿) |
18 | | ssrab2 3650 |
. . . . . . . . . . . . 13
⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉 |
19 | 18 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉) |
20 | 13 | elpw 4114 |
. . . . . . . . . . . 12
⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉) |
21 | 19, 20 | sylibr 223 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉) |
22 | | eleq2 2677 |
. . . . . . . . . . . . . . . 16
⊢ (ran
𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿)) |
23 | 22 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿)) |
24 | | ssrelrn 5237 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒) |
25 | 24 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)) |
26 | 25 | 3ad2ant2 1076 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)) |
27 | 23, 26 | sylbird 249 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)) |
28 | 27 | imp 444 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒) |
29 | | df-ne 2782 |
. . . . . . . . . . . . . 14
⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅) |
30 | | rabn0 3912 |
. . . . . . . . . . . . . 14
⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒) |
31 | 29, 30 | bitr3i 265 |
. . . . . . . . . . . . 13
⊢ (¬
{𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒) |
32 | 28, 31 | sylibr 223 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅) |
33 | 13 | elsn 4140 |
. . . . . . . . . . . 12
⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅) |
34 | 32, 33 | sylnibr 318 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅}) |
35 | 21, 34 | eldifd 3551 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅})) |
36 | 35, 14 | fmptd 6292 |
. . . . . . . . 9
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅})) |
37 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})) |
38 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿) |
39 | 37, 38 | feq12d 5946 |
. . . . . . . . 9
⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))) |
40 | 36, 39 | syl5ibr 235 |
. . . . . . . 8
⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
41 | 17, 40 | mpdan 699 |
. . . . . . 7
⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
42 | 41 | imp 444 |
. . . . . 6
⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
43 | 8, 42 | syl6bir 243 |
. . . . 5
⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
44 | 43 | expdimp 452 |
. . . 4
⊢ ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
45 | 44 | impcom 445 |
. . 3
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
46 | | incistruhgr.e |
. . . . . 6
⊢ 𝐸 = (iEdg‘𝐺) |
47 | 10, 46 | isuhgr 25726 |
. . . . 5
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
48 | 47 | 3ad2ant1 1075 |
. . . 4
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
49 | 48 | adantr 480 |
. . 3
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
50 | 45, 49 | mpbird 246 |
. 2
⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph ) |
51 | 50 | ex 449 |
1
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph )) |