Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > upgruhgr | Structured version Visualization version GIF version |
Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.) |
Ref | Expression |
---|---|
upgruhgr | ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2738 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | upgrf 27456 | . . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
4 | ssrab2 4013 | . . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
5 | fss 6617 | . . 3 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) | |
6 | 3, 4, 5 | sylancl 586 | . 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 1, 2 | isuhgr 27430 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
8 | 6, 7 | mpbird 256 | 1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {crab 3068 ∖ cdif 3884 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 ≤ cle 11010 2c2 12028 ♯chash 14044 Vtxcvtx 27366 iEdgciedg 27367 UHGraphcuhgr 27426 UPGraphcupgr 27450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-uhgr 27428 df-upgr 27452 |
This theorem is referenced by: umgruhgr 27474 upgrle2 27475 edglnl 27513 numedglnl 27514 usgruhgr 27553 subupgr 27654 upgrspan 27660 upgrreslem 27671 upgrres 27673 finsumvtxdg2ssteplem1 27912 finsumvtxdg2size 27917 upgrewlkle2 27973 upgredginwlk 28003 wlkiswwlks1 28232 wlkiswwlksupgr2 28242 eulerpathpr 28604 eulercrct 28606 upgracycumgr 33115 isomuspgrlem2c 45282 |
Copyright terms: Public domain | W3C validator |