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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 2610 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | upgrf 25753 | . . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
4 | ssrab2 3650 | . . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
5 | fss 5969 | . . 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 693 | . 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 1, 2 | isuhgr 25726 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
8 | 6, 7 | mpbird 246 | 1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-uhgr 25724 df-upgr 25749 |
This theorem is referenced by: umgruhgr 25770 upgrle2 25771 usgruhgr 40413 subupgr 40511 upgrspan 40517 upgrewlkle2 40808 upgredginwlk 40840 1wlkiswwlks1 41064 1wlkiswwlksupgr2 41074 eulerpathpr 41408 eulercrct 41410 |
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