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Mirrors > Home > MPE Home > Th. List > p1evtxdeqlem | Structured version Visualization version GIF version |
Description: Lemma for p1evtxdeq 29279 and p1evtxdp1 29280. (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 6899 | . . . 4 β’ π β V |
4 | snex 5424 | . . . 4 β’ {β¨πΎ, πΈβ©} β V | |
5 | 3, 4 | pm3.2i 470 | . . 3 β’ (π β V β§ {β¨πΎ, πΈβ©} β V) |
6 | opiedgfv 28775 | . . . 4 β’ ((π β V β§ {β¨πΎ, πΈβ©} β V) β (iEdgββ¨π, {β¨πΎ, πΈβ©}β©) = {β¨πΎ, πΈβ©}) | |
7 | 6 | eqcomd 2732 | . . 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 28772 | . . 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 6206 | . . . . 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 4207 | . . 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 3041 | . . . . 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 4710 | . . . 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 2766 | . 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 6593 | . . 3 β’ ((πΎ β π β§ πΈ β π) β Fun {β¨πΎ, πΈβ©}) | |
25 | 23, 12, 24 | syl2anc 583 | . 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 29247 | 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 395 = wceq 1533 β wcel 2098 β wnel 3040 Vcvv 3468 βͺ cun 3941 β© cin 3942 β c0 4317 {csn 4623 β¨cop 4629 dom cdm 5669 Fun wfun 6531 βcfv 6537 (class class class)co 7405 +π cxad 13096 Vtxcvtx 28764 iEdgciedg 28765 VtxDegcvtxdg 29231 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 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 13099 df-hash 14296 df-vtx 28766 df-iedg 28767 df-vtxdg 29232 |
This theorem is referenced by: p1evtxdeq 29279 p1evtxdp1 29280 |
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