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| Mirrors > Home > MPE Home > Th. List > vtxdusgrfvedg | Structured version Visualization version GIF version | ||
| Description: The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.) |
| Ref | Expression |
|---|---|
| vtxdushgrfvedg.v | β’ π = (VtxβπΊ) |
| vtxdushgrfvedg.e | β’ πΈ = (EdgβπΊ) |
| vtxdushgrfvedg.d | β’ π· = (VtxDegβπΊ) |
| Ref | Expression |
|---|---|
| vtxdusgrfvedg | β’ ((πΊ β USGraph β§ π β π) β (π·βπ) = (β―β{π β πΈ β£ π β π})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxdushgrfvedg.v | . . 3 β’ π = (VtxβπΊ) | |
| 2 | eqid 2727 | . . 3 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 3 | eqid 2727 | . . 3 β’ dom (iEdgβπΊ) = dom (iEdgβπΊ) | |
| 4 | vtxdushgrfvedg.d | . . 3 β’ π· = (VtxDegβπΊ) | |
| 5 | 1, 2, 3, 4 | vtxdusgrval 29288 | . 2 β’ ((πΊ β USGraph β§ π β π) β (π·βπ) = (β―β{π β dom (iEdgβπΊ) β£ π β ((iEdgβπΊ)βπ)})) |
| 6 | usgruspgr 28980 | . . . 4 β’ (πΊ β USGraph β πΊ β USPGraph) | |
| 7 | uspgrushgr 28977 | . . . 4 β’ (πΊ β USPGraph β πΊ β USHGraph) | |
| 8 | 6, 7 | syl 17 | . . 3 β’ (πΊ β USGraph β πΊ β USHGraph) |
| 9 | vtxdushgrfvedg.e | . . . 4 β’ πΈ = (EdgβπΊ) | |
| 10 | 1, 9 | vtxdushgrfvedglem 29290 | . . 3 β’ ((πΊ β USHGraph β§ π β π) β (β―β{π β dom (iEdgβπΊ) β£ π β ((iEdgβπΊ)βπ)}) = (β―β{π β πΈ β£ π β π})) |
| 11 | 8, 10 | sylan 579 | . 2 β’ ((πΊ β USGraph β§ π β π) β (β―β{π β dom (iEdgβπΊ) β£ π β ((iEdgβπΊ)βπ)}) = (β―β{π β πΈ β£ π β π})) |
| 12 | 5, 11 | eqtrd 2767 | 1 β’ ((πΊ β USGraph β§ π β π) β (π·βπ) = (β―β{π β πΈ β£ π β π})) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 {crab 3427 dom cdm 5672 βcfv 6542 β―chash 14313 Vtxcvtx 28796 iEdgciedg 28797 Edgcedg 28847 USHGraphcushgr 28857 USPGraphcuspgr 28948 USGraphcusgr 28949 VtxDegcvtxdg 29266 |
| 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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-xadd 13117 df-fz 13509 df-hash 14314 df-edg 28848 df-uhgr 28858 df-ushgr 28859 df-upgr 28882 df-umgr 28883 df-uspgr 28950 df-usgr 28951 df-vtxdg 29267 |
| This theorem is referenced by: vtxdusgr0edgnelALT 29297 hashnbusgrvd 29329 |
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