Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uhgr0vb | Structured version Visualization version GIF version |
Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgr0vb | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2739 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 27010 | . . 3 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | pweq 4505 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅) | |
5 | 4 | difeq1d 4013 | . . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅})) |
6 | pw0 4701 | . . . . . . . . 9 ⊢ 𝒫 ∅ = {∅} | |
7 | 6 | difeq1i 4010 | . . . . . . . 8 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
8 | difid 4260 | . . . . . . . 8 ⊢ ({∅} ∖ {∅}) = ∅ | |
9 | 7, 8 | eqtri 2762 | . . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
10 | 5, 9 | eqtrdi 2790 | . . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
12 | 11 | feq3d 6492 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)) |
13 | f00 6561 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅)) | |
14 | 13 | simplbi 501 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅) |
15 | 12, 14 | syl6bi 256 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺) = ∅)) |
16 | 3, 15 | syl5 34 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)) |
17 | simpl 486 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
18 | simpr 488 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
19 | 17, 18 | uhgr0e 27019 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph) |
20 | 19 | ex 416 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
21 | 20 | adantr 484 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
22 | 16, 21 | impbid 215 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3841 ∅c0 4212 𝒫 cpw 4489 {csn 4517 dom cdm 5526 ⟶wf 6336 ‘cfv 6340 Vtxcvtx 26944 iEdgciedg 26945 UHGraphcuhgr 27004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-uhgr 27006 |
This theorem is referenced by: usgr0vb 27182 uhgr0v0e 27183 0uhgrsubgr 27224 finsumvtxdg2size 27495 0uhgrrusgr 27523 frgr0v 28202 frgruhgr0v 28204 |
Copyright terms: Public domain | W3C validator |