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Mirrors > Home > MPE Home > Th. List > uspgrupgr | Structured version Visualization version GIF version |
Description: A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
uspgrupgr | ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2726 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 29088 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | f1f 6798 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
5 | 3, 4 | biimtrdi 252 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
6 | 1, 2 | isupgr 29020 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
7 | 5, 6 | sylibrd 258 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)) |
8 | 7 | pm2.43i 52 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 {crab 3419 ∖ cdif 3944 ∅c0 4325 𝒫 cpw 4607 {csn 4633 class class class wbr 5153 dom cdm 5682 ⟶wf 6550 –1-1→wf1 6551 ‘cfv 6554 ≤ cle 11299 2c2 12319 ♯chash 14347 Vtxcvtx 28932 iEdgciedg 28933 UPGraphcupgr 29016 USPGraphcuspgr 29084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5311 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fv 6562 df-upgr 29018 df-uspgr 29086 |
This theorem is referenced by: uspgrupgrushgr 29115 uspgruhgr 29120 usgrupgr 29121 uspgrun 29124 uspgrunop 29125 uspgredg2vtxeu 29156 1loopgrnb0 29439 uspgr2wlkeq 29583 uspgrn2crct 29742 wlkiswwlks2 29809 wlkiswwlks 29810 wlklnwwlkn 29818 clwlkclwwlk 29935 wlk2v2e 30090 isuspgrim0 47451 isuspgrimlem 47453 uspgropssxp 47521 uspgrsprf 47523 |
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