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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 2731 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2731 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 28676 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | f1f 6788 | . . . 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 | syl6bi 252 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
6 | 1, 2 | isupgr 28608 | . . 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 2105 {crab 3431 ∖ cdif 3946 ∅c0 4323 𝒫 cpw 4603 {csn 4629 class class class wbr 5149 dom cdm 5677 ⟶wf 6540 –1-1→wf1 6541 ‘cfv 6544 ≤ cle 11254 2c2 12272 ♯chash 14295 Vtxcvtx 28520 iEdgciedg 28521 UPGraphcupgr 28604 USPGraphcuspgr 28672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rab 3432 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fv 6552 df-upgr 28606 df-uspgr 28674 |
This theorem is referenced by: uspgrupgrushgr 28701 usgrupgr 28706 uspgrun 28709 uspgrunop 28710 uspgredg2vtxeu 28741 1loopgrnb0 29023 uspgr2wlkeq 29167 uspgrn2crct 29326 wlkiswwlks2 29393 wlkiswwlks 29394 wlklnwwlkn 29402 clwlkclwwlk 29519 wlk2v2e 29674 isomuspgrlem1 46795 isomuspgrlem2b 46797 isomuspgrlem2c 46798 isomuspgrlem2d 46799 uspgropssxp 46822 uspgrsprf 46824 |
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