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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 2818 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2818 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isusgr 26865 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
4 | 2re 11699 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
5 | 4 | eqlei2 10739 | . . . . . . 7 ⊢ ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑥) = 2 → (♯‘𝑥) ≤ 2)) |
7 | 6 | ss2rabi 4050 | . . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
8 | f1ss 6573 | . . . . 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 687 | . . . 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 | syl6bi 254 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
11 | 1, 2 | isuspgr 26864 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
12 | 10, 11 | sylibrd 260 | . 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 1528 ∈ wcel 2105 {crab 3139 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 class class class wbr 5057 dom cdm 5548 –1-1→wf1 6345 ‘cfv 6348 ≤ cle 10664 2c2 11680 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 USPGraphcuspgr 26860 USGraphcusgr 26861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-2 11688 df-uspgr 26862 df-usgr 26863 |
This theorem is referenced by: usgrumgruspgr 26892 usgruspgrb 26893 usgrupgr 26894 usgrislfuspgr 26896 usgredg2vtxeu 26930 usgredgedg 26939 usgredgleord 26942 vtxdusgrfvedg 27200 usgrn2cycl 27514 wlksnfi 27613 rusgrnumwwlk 27681 rusgrnumwlkg 27683 clwlksndivn 27792 clwlknon2num 28074 numclwlk1lem2 28076 |
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