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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 2736 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2736 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 27811 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | f1f 6721 | . . . 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 27743 | . . 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 3403 ∖ cdif 3895 ∅c0 4269 𝒫 cpw 4547 {csn 4573 class class class wbr 5092 dom cdm 5620 ⟶wf 6475 –1-1→wf1 6476 ‘cfv 6479 ≤ cle 11111 2c2 12129 ♯chash 14145 Vtxcvtx 27655 iEdgciedg 27656 UPGraphcupgr 27739 USPGraphcuspgr 27807 |
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 2707 ax-nul 5250 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fv 6487 df-upgr 27741 df-uspgr 27809 |
This theorem is referenced by: uspgrupgrushgr 27836 usgrupgr 27841 uspgrun 27844 uspgrunop 27845 uspgredg2vtxeu 27876 1loopgrnb0 28158 uspgr2wlkeq 28302 uspgrn2crct 28461 wlkiswwlks2 28528 wlkiswwlks 28529 wlklnwwlkn 28537 clwlkclwwlk 28654 wlk2v2e 28809 isomuspgrlem1 45638 isomuspgrlem2b 45640 isomuspgrlem2c 45641 isomuspgrlem2d 45642 uspgropssxp 45665 uspgrsprf 45667 |
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