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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 2821 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | ushgrf 26842 | . . 3 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | f1f 6570 | . . 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 26839 | . 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
7 | 5, 6 | mpbird 259 | 1 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4561 dom cdm 5550 ⟶wf 6346 –1-1→wf1 6347 ‘cfv 6350 Vtxcvtx 26775 iEdgciedg 26776 UHGraphcuhgr 26835 USHGraphcushgr 26836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fv 6358 df-uhgr 26837 df-ushgr 26838 |
This theorem is referenced by: ushgrun 26855 ushgrunop 26856 ushgredgedg 27005 ushgredgedgloop 27007 ushrisomgr 43999 |
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