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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 29109 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | anim1ci 616 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → (𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph)) |
3 | eqid 2736 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2736 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | acycgrislfgr 35135 | . . 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 29130 | . . 3 ⊢ (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)})) |
8 | 7 | biimpri 228 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) → 𝐺 ∈ UMGraph) |
9 | 6, 8 | syldan 591 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {crab 3435 𝒫 cpw 4598 class class class wbr 5141 dom cdm 5683 ⟶wf 6555 ‘cfv 6559 ≤ cle 11292 2c2 12317 ♯chash 14365 Vtxcvtx 29003 iEdgciedg 29004 UHGraphcuhgr 29063 UPGraphcupgr 29087 UMGraphcumgr 29088 AcyclicGraphcacycgr 35125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-oadd 8506 df-er 8741 df-map 8864 df-pm 8865 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-dju 9937 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12875 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-edg 29055 df-uhgr 29065 df-upgr 29089 df-umgr 29090 df-wlks 29607 df-wlkson 29608 df-trls 29700 df-trlson 29701 df-pths 29724 df-pthson 29726 df-cycls 29797 df-acycgr 35126 |
This theorem is referenced by: upgracycusgr 35138 |
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