Step | Hyp | Ref
| Expression |
1 | | df-conngr 41354 |
. . 3
⊢ ConnGraph
= {𝑔 ∣
[(Vtx‘𝑔) /
𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} |
2 | 1 | eleq2i 2680 |
. 2
⊢ (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}) |
3 | | fvex 6113 |
. . . . . 6
⊢
(Vtx‘𝑔) ∈
V |
4 | | id 22 |
. . . . . . 7
⊢ (𝑣 = (Vtx‘𝑔) → 𝑣 = (Vtx‘𝑔)) |
5 | | raleq 3115 |
. . . . . . 7
⊢ (𝑣 = (Vtx‘𝑔) → (∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝)) |
6 | 4, 5 | raleqbidv 3129 |
. . . . . 6
⊢ (𝑣 = (Vtx‘𝑔) → (∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝)) |
7 | 3, 6 | sbcie 3437 |
. . . . 5
⊢
([(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝) |
8 | 7 | abbii 2726 |
. . . 4
⊢ {𝑔 ∣
[(Vtx‘𝑔) /
𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} |
9 | 8 | eleq2i 2680 |
. . 3
⊢ (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ 𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}) |
10 | | fveq2 6103 |
. . . . . 6
⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺)) |
11 | | isconngr.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
12 | 10, 11 | syl6eqr 2662 |
. . . . 5
⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉) |
13 | | fveq2 6103 |
. . . . . . . . 9
⊢ (ℎ = 𝐺 → (PathsOn‘ℎ) = (PathsOn‘𝐺)) |
14 | 13 | oveqd 6566 |
. . . . . . . 8
⊢ (ℎ = 𝐺 → (𝑘(PathsOn‘ℎ)𝑛) = (𝑘(PathsOn‘𝐺)𝑛)) |
15 | 14 | breqd 4594 |
. . . . . . 7
⊢ (ℎ = 𝐺 → (𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
16 | 15 | 2exbidv 1839 |
. . . . . 6
⊢ (ℎ = 𝐺 → (∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
17 | 12, 16 | raleqbidv 3129 |
. . . . 5
⊢ (ℎ = 𝐺 → (∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
18 | 12, 17 | raleqbidv 3129 |
. . . 4
⊢ (ℎ = 𝐺 → (∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
19 | | fveq2 6103 |
. . . . . 6
⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ)) |
20 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑔 = ℎ → (PathsOn‘𝑔) = (PathsOn‘ℎ)) |
21 | 20 | oveqd 6566 |
. . . . . . . . 9
⊢ (𝑔 = ℎ → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘ℎ)𝑛)) |
22 | 21 | breqd 4594 |
. . . . . . . 8
⊢ (𝑔 = ℎ → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝)) |
23 | 22 | 2exbidv 1839 |
. . . . . . 7
⊢ (𝑔 = ℎ → (∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝)) |
24 | 19, 23 | raleqbidv 3129 |
. . . . . 6
⊢ (𝑔 = ℎ → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝)) |
25 | 19, 24 | raleqbidv 3129 |
. . . . 5
⊢ (𝑔 = ℎ → (∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝)) |
26 | 25 | cbvabv 2734 |
. . . 4
⊢ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = {ℎ ∣ ∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝} |
27 | 18, 26 | elab2g 3322 |
. . 3
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
28 | 9, 27 | syl5bb 271 |
. 2
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |
29 | 2, 28 | syl5bb 271 |
1
⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) |