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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 6789 | . . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) | |
| 2 | dm0 5931 | . . . 4 ⊢ dom ∅ = ∅ | |
| 3 | 2 | feq2i 6728 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) | 
| 4 | 1, 3 | mpbir 231 | . 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) | 
| 5 | uhgr0e.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 7 | eqid 2737 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 8 | 6, 7 | isuhgr 29077 | . . . 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 5914 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
| 13 | 11, 12 | feq12d 6724 | . . . 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 279 | . 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) | 
| 16 | 4, 15 | mpbiri 258 | 1 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = 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: uhgr0vb 29089 | 
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