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Mirrors > Home > MPE Home > Th. List > ushgruhgr | Structured version Visualization version GIF version |
Description: An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
ushgruhgr | ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2734 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | ushgrf 27126 | . . 3 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | f1f 6604 | . . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
6 | 1, 2 | isuhgr 27123 | . 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
7 | 5, 6 | mpbird 260 | 1 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∖ cdif 3854 ∅c0 4227 𝒫 cpw 4503 {csn 4531 dom cdm 5540 ⟶wf 6365 –1-1→wf1 6366 ‘cfv 6369 Vtxcvtx 27059 iEdgciedg 27060 UHGraphcuhgr 27119 USHGraphcushgr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-nul 5188 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fv 6377 df-uhgr 27121 df-ushgr 27122 |
This theorem is referenced by: ushgrun 27139 ushgrunop 27140 ushgredgedg 27289 ushgredgedgloop 27291 ushrisomgr 44920 |
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