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Mirrors > Home > MPE Home > Th. List > p1evtxdeqlem | Structured version Visualization version GIF version |
Description: Lemma for p1evtxdeq 29371 and p1evtxdp1 29372. (Contributed by AV, 3-Mar-2021.) |
Ref | Expression |
---|---|
p1evtxdeq.v | β’ π = (VtxβπΊ) |
p1evtxdeq.i | β’ πΌ = (iEdgβπΊ) |
p1evtxdeq.f | β’ (π β Fun πΌ) |
p1evtxdeq.fv | β’ (π β (VtxβπΉ) = π) |
p1evtxdeq.fi | β’ (π β (iEdgβπΉ) = (πΌ βͺ {β¨πΎ, πΈβ©})) |
p1evtxdeq.k | β’ (π β πΎ β π) |
p1evtxdeq.d | β’ (π β πΎ β dom πΌ) |
p1evtxdeq.u | β’ (π β π β π) |
p1evtxdeq.e | β’ (π β πΈ β π) |
Ref | Expression |
---|---|
p1evtxdeqlem | β’ (π β ((VtxDegβπΉ)βπ) = (((VtxDegβπΊ)βπ) +π ((VtxDegββ¨π, {β¨πΎ, πΈβ©}β©)βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p1evtxdeq.i | . 2 β’ πΌ = (iEdgβπΊ) | |
2 | p1evtxdeq.v | . . . . 5 β’ π = (VtxβπΊ) | |
3 | 2 | fvexi 6906 | . . . 4 β’ π β V |
4 | snex 5427 | . . . 4 β’ {β¨πΎ, πΈβ©} β V | |
5 | 3, 4 | pm3.2i 469 | . . 3 β’ (π β V β§ {β¨πΎ, πΈβ©} β V) |
6 | opiedgfv 28864 | . . . 4 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β (iEdgββ¨π, {β¨πΎ, πΈβ©}β©) = {β¨πΎ, πΈβ©}) | |
7 | 6 | eqcomd 2731 | . . 3 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β {β¨πΎ, πΈβ©} = (iEdgββ¨π, {β¨πΎ, πΈβ©}β©)) |
8 | 5, 7 | ax-mp 5 | . 2 β’ {β¨πΎ, πΈβ©} = (iEdgββ¨π, {β¨πΎ, πΈβ©}β©) |
9 | opvtxfv 28861 | . . 3 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β (Vtxββ¨π, {β¨πΎ, πΈβ©}β©) = π) | |
10 | 5, 9 | mp1i 13 | . 2 β’ (π β (Vtxββ¨π, {β¨πΎ, πΈβ©}β©) = π) |
11 | p1evtxdeq.fv | . 2 β’ (π β (VtxβπΉ) = π) | |
12 | p1evtxdeq.e | . . . . 5 β’ (π β πΈ β π) | |
13 | dmsnopg 6212 | . . . . 5 β’ (πΈ β π β dom {β¨πΎ, πΈβ©} = {πΎ}) | |
14 | 12, 13 | syl 17 | . . . 4 β’ (π β dom {β¨πΎ, πΈβ©} = {πΎ}) |
15 | 14 | ineq2d 4206 | . . 3 β’ (π β (dom πΌ β© dom {β¨πΎ, πΈβ©}) = (dom πΌ β© {πΎ})) |
16 | p1evtxdeq.d | . . . . 5 β’ (π β πΎ β dom πΌ) | |
17 | df-nel 3037 | . . . . 5 β’ (πΎ β dom πΌ β Β¬ πΎ β dom πΌ) | |
18 | 16, 17 | sylib 217 | . . . 4 β’ (π β Β¬ πΎ β dom πΌ) |
19 | disjsn 4711 | . . . 4 β’ ((dom πΌ β© {πΎ}) = β β Β¬ πΎ β dom πΌ) | |
20 | 18, 19 | sylibr 233 | . . 3 β’ (π β (dom πΌ β© {πΎ}) = β ) |
21 | 15, 20 | eqtrd 2765 | . 2 β’ (π β (dom πΌ β© dom {β¨πΎ, πΈβ©}) = β ) |
22 | p1evtxdeq.f | . 2 β’ (π β Fun πΌ) | |
23 | p1evtxdeq.k | . . 3 β’ (π β πΎ β π) | |
24 | funsng 6599 | . . 3 β’ ((πΎ β π β§ πΈ β π) β Fun {β¨πΎ, πΈβ©}) | |
25 | 23, 12, 24 | syl2anc 582 | . 2 β’ (π β Fun {β¨πΎ, πΈβ©}) |
26 | p1evtxdeq.u | . 2 β’ (π β π β π) | |
27 | p1evtxdeq.fi | . 2 β’ (π β (iEdgβπΉ) = (πΌ βͺ {β¨πΎ, πΈβ©})) | |
28 | 1, 8, 2, 10, 11, 21, 22, 25, 26, 27 | vtxdun 29339 | 1 β’ (π β ((VtxDegβπΉ)βπ) = (((VtxDegβπΊ)βπ) +π ((VtxDegββ¨π, {β¨πΎ, πΈβ©}β©)βπ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wnel 3036 Vcvv 3463 βͺ cun 3937 β© cin 3938 β c0 4318 {csn 4624 β¨cop 4630 dom cdm 5672 Fun wfun 6537 βcfv 6543 (class class class)co 7416 +π cxad 13122 Vtxcvtx 28853 iEdgciedg 28854 VtxDegcvtxdg 29323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-xadd 13125 df-hash 14322 df-vtx 28855 df-iedg 28856 df-vtxdg 29324 |
This theorem is referenced by: p1evtxdeq 29371 p1evtxdp1 29372 |
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