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Mirrors > Home > MPE Home > Th. List > eupthvdres | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2 27946: 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 5348 | . . . 4 ⊢ 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V | |
4 | 2, 3 | eqeltri 2909 | . . 3 ⊢ 𝐻 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
6 | 2 | fveq2i 6667 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | fvexi 6678 | . . . . . . 7 ⊢ 𝑉 ∈ V |
9 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
10 | 9 | fvexi 6678 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
11 | 10 | resex 5893 | . . . . . . 7 ⊢ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V |
12 | 8, 11 | pm3.2i 471 | . . . . . 6 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V)) |
14 | opvtxfv 26717 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) |
16 | 6, 15 | syl5eq 2868 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
17 | 16, 7 | syl6eq 2872 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) |
18 | 2 | fveq2i 6667 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
19 | opiedgfv 26720 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) | |
20 | 13, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
21 | 18, 20 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
22 | eupthvdres.p | . . . . . . 7 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
23 | 9 | eupthf1o 27911 | . . . . . . 7 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
25 | f1ofo 6616 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
26 | foima 6589 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | |
27 | 24, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) |
28 | 27 | reseq2d 5847 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
29 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
30 | 29 | funfnd 6380 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
31 | fnresdm 6460 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼) | |
32 | 30, 31 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼) |
33 | 21, 28, 32 | 3eqtrd 2860 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) |
34 | 33, 9 | syl6eq 2872 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) |
35 | 1, 5, 17, 34 | vtxdeqd 27187 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3495 〈cop 4565 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 “ cima 5552 Fun wfun 6343 Fn wfn 6344 –onto→wfo 6347 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7145 0cc0 10526 ..^cfzo 13023 ♯chash 13680 Vtxcvtx 26709 iEdgciedg 26710 VtxDegcvtxdg 27175 EulerPathsceupth 27904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-vtx 26711 df-iedg 26712 df-vtxdg 27176 df-wlks 27309 df-trls 27402 df-eupth 27905 |
This theorem is referenced by: eupth2 27946 |
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