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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 2821 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | umgrf 26883 | . . 3 ⊢ (𝐺 ∈ UMGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
4 | isacycgr 32392 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
5 | 4 | biimpa 479 | . . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
6 | 2 | umgr2cycl 32388 | . . . . . . . 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 11742 | . . . . . . . . . . . 12 ⊢ 2 ≠ 0 | |
8 | neeq1 3078 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) ≠ 0 ↔ 2 ≠ 0)) | |
9 | 7, 8 | mpbiri 260 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 2 → (♯‘𝑓) ≠ 0) |
10 | hasheq0 13725 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
11 | 10 | elv 3499 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
12 | 11 | necon3bii 3068 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
13 | 9, 12 | sylib 220 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 2 → 𝑓 ≠ ∅) |
14 | 13 | anim2i 618 | . . . . . . . . 9 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
15 | 14 | 2eximi 1836 | . . . . . . . 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 415 | . . . . . 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 483 | . . . 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 32353 | . . . 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 230 | . . 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 683 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
24 | 1, 2 | isusgrs 26941 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})) |
25 | 24 | biimprd 250 | . . 3 ⊢ (𝐺 ∈ UMGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} → 𝐺 ∈ USGraph)) |
26 | 25 | adantr 483 | . 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 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ∅c0 4291 𝒫 cpw 4539 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 –1-1→wf1 6352 ‘cfv 6355 0cc0 10537 2c2 11693 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 UMGraphcumgr 26866 USGraphcusgr 26934 Cyclesccycls 27566 AcyclicGraphcacycgr 32389 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-umgr 26868 df-usgr 26936 df-wlks 27381 df-trls 27474 df-pths 27497 df-cycls 27568 df-acycgr 32390 |
This theorem is referenced by: upgracycusgr 32402 |
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