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Mirrors > Home > MPE Home > Th. List > Mathboxes > upgracycumgr | Structured version Visualization version GIF version |
Description: An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.) |
Ref | Expression |
---|---|
upgracycumgr | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgruhgr 28954 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | anim1ci 614 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → (𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph)) |
3 | eqid 2725 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2725 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | acycgrislfgr 34815 | . . 3 ⊢ ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
6 | 2, 5 | syl 17 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
7 | 3, 4 | umgrislfupgr 28975 | . . 3 ⊢ (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)})) |
8 | 7 | biimpri 227 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) → 𝐺 ∈ UMGraph) |
9 | 6, 8 | syldan 589 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 {crab 3419 𝒫 cpw 4599 class class class wbr 5144 dom cdm 5673 ⟶wf 6539 ‘cfv 6543 ≤ cle 11274 2c2 12292 ♯chash 14316 Vtxcvtx 28848 iEdgciedg 28849 UHGraphcuhgr 28908 UPGraphcupgr 28932 UMGraphcumgr 28933 AcyclicGraphcacycgr 34805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-s2 14826 df-edg 28900 df-uhgr 28910 df-upgr 28934 df-umgr 28935 df-wlks 29452 df-wlkson 29453 df-trls 29545 df-trlson 29546 df-pths 29569 df-pthson 29571 df-cycls 29640 df-acycgr 34806 |
This theorem is referenced by: upgracycusgr 34818 |
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