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Mirrors > Home > MPE Home > Th. List > Mathboxes > umgracycusgr | Structured version Visualization version GIF version |
Description: An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.) |
Ref | Expression |
---|---|
umgracycusgr | ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | umgrf 26891 | . . 3 ⊢ (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
4 | isacycgr 32505 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
5 | 4 | biimpa 480 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
6 | 2 | umgr2cycl 32501 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2)) |
7 | 2ne0 11729 | . . . . . . . . . . . 12 ⊢ 2 ≠ 0 | |
8 | neeq1 3049 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) ≠ 0 ↔ 2 ≠ 0)) | |
9 | 7, 8 | mpbiri 261 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 2 → (♯‘𝑓) ≠ 0) |
10 | hasheq0 13720 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
11 | 10 | elv 3446 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
12 | 11 | necon3bii 3039 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
13 | 9, 12 | sylib 221 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 2 → 𝑓 ≠ ∅) |
14 | 13 | anim2i 619 | . . . . . . . . 9 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
15 | 14 | 2eximi 1837 | . . . . . . . 8 ⊢ (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
16 | 6, 15 | syl 17 | . . . . . . 7 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
17 | 16 | ex 416 | . . . . . 6 ⊢ (𝐺 ∈ UMGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
18 | 17 | con3d 155 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘))) |
19 | 18 | adantr 484 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘))) |
20 | 5, 19 | mpd 15 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → ¬ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘)) |
21 | dff15 32463 | . . . 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} ∧ ¬ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘))) | |
22 | 21 | biimpri 231 | . . 3 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} ∧ ¬ ∃𝑗 ∈ dom (iEdg‘𝐺)∃𝑘 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑗) = ((iEdg‘𝐺)‘𝑘) ∧ 𝑗 ≠ 𝑘)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
23 | 3, 20, 22 | syl2an2r 684 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
24 | 1, 2 | isusgrs 26949 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})) |
25 | 24 | biimprd 251 | . . 3 ⊢ (𝐺 ∈ UMGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} → 𝐺 ∈ USGraph)) |
26 | 25 | adantr 484 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} → 𝐺 ∈ USGraph)) |
27 | 23, 26 | mpd 15 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 Vcvv 3441 ∅c0 4243 𝒫 cpw 4497 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 –1-1→wf1 6321 ‘cfv 6324 0cc0 10526 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 UMGraphcumgr 26874 USGraphcusgr 26942 Cyclesccycls 27574 AcyclicGraphcacycgr 32502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-edg 26841 df-uhgr 26851 df-upgr 26875 df-umgr 26876 df-usgr 26944 df-wlks 27389 df-trls 27482 df-pths 27505 df-cycls 27576 df-acycgr 32503 |
This theorem is referenced by: upgracycusgr 32515 |
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