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Mirrors > Home > MPE Home > Th. List > usgrumgr | Structured version Visualization version GIF version |
Description: A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
usgrumgr | ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2735 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | usgrfs 29189 | . . 3 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
4 | f1f 6805 | . . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
6 | 1, 2 | isumgrs 29128 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})) |
7 | 5, 6 | mpbird 257 | 1 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 𝒫 cpw 4605 dom cdm 5689 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 UMGraphcumgr 29113 USGraphcusgr 29181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 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 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-umgr 29115 df-usgr 29183 |
This theorem is referenced by: usgrumgruspgr 29214 usgrun 29222 usgrunop 29223 usgredg2 29224 usgredgprv 29226 usgrpredgv 29229 usgredg 29231 usgrnloopv 29232 usgrnloop 29234 usgrnloop0 29236 usgredgne 29238 usgr2edg 29242 usgr2edg1 29244 subusgr 29321 usgrres 29340 usgrres1 29347 nbusgrvtx 29380 nbusgr 29381 vtxdusgrval 29520 usgr2wspthons3 29994 fusgrhashclwwlkn 30108 3cyclfrgr 30317 vdgn0frgrv2 30324 frgrncvvdeqlem2 30329 frgr2wwlkeu 30356 frgr2wwlkeqm 30360 fusgr2wsp2nb 30363 usgrgt2cycl 35115 cusgr3cyclex 35121 |
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