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Mirrors > Home > MPE Home > Th. List > usgruspgr | Structured version Visualization version GIF version |
Description: A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
usgruspgr | ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2735 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isusgr 29185 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
4 | 2re 12338 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
5 | 4 | eqlei2 11370 | . . . . . . 7 ⊢ ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)) |
7 | 6 | ss2rabi 4087 | . . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
8 | f1ss 6810 | . . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
9 | 7, 8 | mpan2 691 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
10 | 3, 9 | biimtrdi 253 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
11 | 1, 2 | isuspgr 29184 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
12 | 10, 11 | sylibrd 259 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)) |
13 | 12 | pm2.43i 52 | 1 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 class class class wbr 5148 dom cdm 5689 –1-1→wf1 6560 ‘cfv 6563 ≤ cle 11294 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 USPGraphcuspgr 29180 USGraphcusgr 29181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-2 12327 df-uspgr 29182 df-usgr 29183 |
This theorem is referenced by: usgrumgruspgr 29214 usgruspgrb 29215 usgrupgr 29217 usgrislfuspgr 29219 usgredg2vtxeu 29253 usgredgedg 29262 usgredgleord 29265 vtxdusgrfvedg 29524 usgrn2cycl 29839 wlksnfi 29937 rusgrnumwwlk 30005 rusgrnumwlkg 30007 clwlksndivn 30115 clwlknon2num 30397 numclwlk1lem2 30399 isubgr3stgr 47878 usgrexmpl12ngrlic 47934 |
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