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Mirrors > Home > MPE Home > Th. List > eupthvdres | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2 27362: The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
Ref | Expression |
---|---|
eupthvdres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
eupthvdres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
eupthvdres.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
eupthvdres.f | ⊢ (𝜑 → Fun 𝐼) |
eupthvdres.p | ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) |
eupthvdres.h | ⊢ 𝐻 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 |
Ref | Expression |
---|---|
eupthvdres | ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupthvdres.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
2 | eupthvdres.h | . . . 4 ⊢ 𝐻 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 | |
3 | opex 5069 | . . . 4 ⊢ 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V | |
4 | 2, 3 | eqeltri 2823 | . . 3 ⊢ 𝐻 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
6 | 2 | fveq2i 6343 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | fvex 6350 | . . . . . . . 8 ⊢ (Vtx‘𝐺) ∈ V | |
9 | 7, 8 | eqeltri 2823 | . . . . . . 7 ⊢ 𝑉 ∈ V |
10 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
11 | fvex 6350 | . . . . . . . . 9 ⊢ (iEdg‘𝐺) ∈ V | |
12 | 10, 11 | eqeltri 2823 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
13 | 12 | resex 5589 | . . . . . . 7 ⊢ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V |
14 | 9, 13 | pm3.2i 470 | . . . . . 6 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V)) |
16 | opvtxfv 26054 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) |
18 | 6, 17 | syl5eq 2794 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
19 | 18, 7 | syl6eq 2798 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) |
20 | 2 | fveq2i 6343 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
21 | opiedgfv 26057 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) | |
22 | 15, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
23 | 20, 22 | syl5eq 2794 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
24 | eupthvdres.p | . . . . . . 7 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
25 | 10 | eupthf1o 27327 | . . . . . . 7 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
27 | f1ofo 6293 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
28 | foima 6269 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | |
29 | 26, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) |
30 | 29 | reseq2d 5539 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
31 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
32 | funfn 6067 | . . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
33 | 31, 32 | sylib 208 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
34 | fnresdm 6149 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼) |
36 | 23, 30, 35 | 3eqtrd 2786 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) |
37 | 36, 10 | syl6eq 2798 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) |
38 | 1, 5, 19, 37 | vtxdeqd 26554 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 Vcvv 3328 〈cop 4315 class class class wbr 4792 dom cdm 5254 ↾ cres 5256 “ cima 5257 Fun wfun 6031 Fn wfn 6032 –onto→wfo 6035 –1-1-onto→wf1o 6036 ‘cfv 6037 (class class class)co 6801 0cc0 10099 ..^cfzo 12630 ♯chash 13282 Vtxcvtx 26044 iEdgciedg 26045 VtxDegcvtxdg 26542 EulerPathsceupth 27320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1051 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-er 7899 df-map 8013 df-pm 8014 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8926 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-fzo 12631 df-hash 13283 df-word 13456 df-vtx 26046 df-iedg 26047 df-vtxdg 26543 df-wlks 26676 df-trls 26770 df-eupth 27321 |
This theorem is referenced by: eupth2 27362 |
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