![]() |
Mathbox for BTernaryTau |
< Previous
Next >
Nearby theorems |
|
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 29115 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | anim1ci 615 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → (𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph)) |
3 | eqid 2733 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2733 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | acycgrislfgr 35098 | . . 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 29136 | . . 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 590 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2104 {crab 3432 𝒫 cpw 4604 class class class wbr 5149 dom cdm 5683 ⟶wf 6554 ‘cfv 6558 ≤ cle 11287 2c2 12312 ♯chash 14355 Vtxcvtx 29009 iEdgciedg 29010 UHGraphcuhgr 29069 UPGraphcupgr 29093 UMGraphcumgr 29094 AcyclicGraphcacycgr 35088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-oadd 8503 df-er 8738 df-map 8861 df-pm 8862 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-n0 12518 df-xnn0 12591 df-z 12605 df-uz 12870 df-fz 13538 df-fzo 13682 df-hash 14356 df-word 14539 df-concat 14595 df-s1 14620 df-s2 14873 df-edg 29061 df-uhgr 29071 df-upgr 29095 df-umgr 29096 df-wlks 29613 df-wlkson 29614 df-trls 29706 df-trlson 29707 df-pths 29730 df-pthson 29732 df-cycls 29801 df-acycgr 35089 |
This theorem is referenced by: upgracycusgr 35101 |
Copyright terms: Public domain | W3C validator |