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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 2799 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2799 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 26388 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | f1f 6316 | . . . 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 245 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
6 | 1, 2 | isupgr 26319 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
7 | 5, 6 | sylibrd 251 | . 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 2157 {crab 3093 ∖ cdif 3766 ∅c0 4115 𝒫 cpw 4349 {csn 4368 class class class wbr 4843 dom cdm 5312 ⟶wf 6097 –1-1→wf1 6098 ‘cfv 6101 ≤ cle 10364 2c2 11368 ♯chash 13370 Vtxcvtx 26231 iEdgciedg 26232 UPGraphcupgr 26315 USPGraphcuspgr 26384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-nul 4983 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fv 6109 df-upgr 26317 df-uspgr 26386 |
This theorem is referenced by: uspgrupgrushgr 26413 usgrupgr 26418 uspgrun 26421 uspgrunop 26422 uspgredg2vtxeu 26453 1loopgrnb0 26752 uspgr2wlkeq 26895 uspgrn2crct 27059 wlkiswwlks2 27132 wlkiswwlks 27133 wlklnwwlkn 27142 wlknwwlksninjOLD 27147 wlknwwlksnsurOLD 27148 wlkwwlkinjOLD 27155 wlkwwlksurOLD 27156 clwlkclwwlk 27295 clwlkclwwlkOLD 27296 wlk2v2e 27501 isomuspgrlem1 42497 isomuspgrlem2b 42499 isomuspgrlem2c 42500 isomuspgrlem2d 42501 uspgropssxp 42551 uspgrsprf 42553 |
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