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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 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 3 | 1, 2 | upgrf 29103 | . . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
| 4 | ssrab2 4080 | . . 3 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
| 5 | fss 6752 | . . 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 29077 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
| 8 | 6, 7 | mpbird 257 | 1 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 ≤ cle 11296 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 UPGraphcupgr 29097 |
| 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-ext 2708 ax-nul 5306 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 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-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 df-upgr 29099 |
| This theorem is referenced by: umgruhgr 29121 upgrle2 29122 edglnl 29160 numedglnl 29161 uspgruhgr 29201 usgruhgr 29203 subupgr 29304 upgrspan 29310 upgrreslem 29321 upgrres 29323 finsumvtxdg2ssteplem1 29563 finsumvtxdg2size 29568 upgrewlkle2 29624 upgredginwlk 29654 wlkiswwlks1 29887 wlkiswwlksupgr2 29897 eulerpathpr 30259 eulercrct 30261 upgracycumgr 35158 isubgrupgr 47856 |
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