| 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 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 3 | 1, 2 | uhgrf 29079 | . . 3 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
| 4 | pweq 4614 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅) | |
| 5 | 4 | difeq1d 4125 | . . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅})) |
| 6 | pw0 4812 | . . . . . . . . 9 ⊢ 𝒫 ∅ = {∅} | |
| 7 | 6 | difeq1i 4122 | . . . . . . . 8 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅}) |
| 8 | difid 4376 | . . . . . . . 8 ⊢ ({∅} ∖ {∅}) = ∅ | |
| 9 | 7, 8 | eqtri 2765 | . . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ∅ |
| 10 | 5, 9 | eqtrdi 2793 | . . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
| 11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅) |
| 12 | 11 | feq3d 6723 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅)) |
| 13 | f00 6790 | . . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅)) | |
| 14 | 13 | simplbi 497 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅) |
| 15 | 12, 14 | biimtrdi 253 | . . 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 482 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
| 18 | simpr 484 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
| 19 | 17, 18 | uhgr0e 29088 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph) |
| 20 | 19 | ex 412 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
| 21 | 20 | adantr 480 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph)) |
| 22 | 16, 21 | impbid 212 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-uhgr 29075 |
| This theorem is referenced by: usgr0vb 29254 uhgr0v0e 29255 0uhgrsubgr 29296 finsumvtxdg2size 29568 0uhgrrusgr 29596 frgr0v 30281 frgruhgr0v 30283 |
| Copyright terms: Public domain | W3C validator |