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| Mirrors > Home > MPE Home > Th. List > vtxdgf | Structured version Visualization version GIF version | ||
| Description: The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
| Ref | Expression |
|---|---|
| vtxdgf.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| vtxdgf | β’ (πΊ β π β (VtxDegβπΊ):πβΆβ0*) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxdgf.v | . . 3 β’ π = (VtxβπΊ) | |
| 2 | eqid 2732 | . . 3 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 3 | eqid 2732 | . . 3 β’ dom (iEdgβπΊ) = dom (iEdgβπΊ) | |
| 4 | 1, 2, 3 | vtxdgfval 28979 | . 2 β’ (πΊ β π β (VtxDegβπΊ) = (π’ β π β¦ ((β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) +π (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}})))) |
| 5 | eqid 2732 | . . . . 5 β’ {π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)} = {π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)} | |
| 6 | fvex 6904 | . . . . . 6 β’ (iEdgβπΊ) β V | |
| 7 | dmexg 7896 | . . . . . 6 β’ ((iEdgβπΊ) β V β dom (iEdgβπΊ) β V) | |
| 8 | 6, 7 | mp1i 13 | . . . . 5 β’ ((πΊ β π β§ π’ β π) β dom (iEdgβπΊ) β V) |
| 9 | 5, 8 | rabexd 5333 | . . . 4 β’ ((πΊ β π β§ π’ β π) β {π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)} β V) |
| 10 | hashxnn0 14303 | . . . 4 β’ ({π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)} β V β (β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) β β0*) | |
| 11 | 9, 10 | syl 17 | . . 3 β’ ((πΊ β π β§ π’ β π) β (β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) β β0*) |
| 12 | eqid 2732 | . . . . 5 β’ {π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}} = {π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}} | |
| 13 | 12, 8 | rabexd 5333 | . . . 4 β’ ((πΊ β π β§ π’ β π) β {π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}} β V) |
| 14 | hashxnn0 14303 | . . . 4 β’ ({π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}} β V β (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}}) β β0*) | |
| 15 | 13, 14 | syl 17 | . . 3 β’ ((πΊ β π β§ π’ β π) β (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}}) β β0*) |
| 16 | xnn0xaddcl 13218 | . . 3 β’ (((β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) β β0* β§ (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}}) β β0*) β ((β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) +π (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}})) β β0*) | |
| 17 | 11, 15, 16 | syl2anc 584 | . 2 β’ ((πΊ β π β§ π’ β π) β ((β―β{π₯ β dom (iEdgβπΊ) β£ π’ β ((iEdgβπΊ)βπ₯)}) +π (β―β{π₯ β dom (iEdgβπΊ) β£ ((iEdgβπΊ)βπ₯) = {π’}})) β β0*) |
| 18 | 4, 17 | fmpt3d 7117 | 1 β’ (πΊ β π β (VtxDegβπΊ):πβΆβ0*) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3432 Vcvv 3474 {csn 4628 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7411 β0*cxnn0 12548 +π cxad 13094 β―chash 14294 Vtxcvtx 28511 iEdgciedg 28512 VtxDegcvtxdg 28977 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-xadd 13097 df-hash 14295 df-vtxdg 28978 |
| This theorem is referenced by: vtxdgelxnn0 28984 vtxdgfisf 28988 |
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