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Mirrors > Home > MPE Home > Th. List > uhgr0e | Structured version Visualization version GIF version |
Description: The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
uhgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
uhgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) |
Ref | Expression |
---|---|
uhgr0e | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6773 | . . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) | |
2 | dm0 5921 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | 2 | feq2i 6710 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | 1, 3 | mpbir 230 | . 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) |
5 | uhgr0e.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
6 | eqid 2730 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
7 | eqid 2730 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
8 | 6, 7 | isuhgr 28585 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
10 | uhgr0e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
11 | id 22 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (iEdg‘𝐺) = ∅) | |
12 | dmeq 5904 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
13 | 11, 12 | feq12d 6706 | . . . 4 ⊢ ((iEdg‘𝐺) = ∅ → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
15 | 9, 14 | bitrd 278 | . 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
16 | 4, 15 | mpbiri 257 | 1 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 ∖ cdif 3946 ∅c0 4323 𝒫 cpw 4603 {csn 4629 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 Vtxcvtx 28521 iEdgciedg 28522 UHGraphcuhgr 28581 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-uhgr 28583 |
This theorem is referenced by: uhgr0vb 28597 |
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