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Mirrors > Home > MPE Home > Th. List > eupthvdres | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2 30126: 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 5466 | . . . 4 ⊢ 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V | |
4 | 2, 3 | eqeltri 2821 | . . 3 ⊢ 𝐻 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
6 | 2 | fveq2i 6899 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | fvexi 6910 | . . . . . . 7 ⊢ 𝑉 ∈ V |
9 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
10 | 9 | fvexi 6910 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
11 | 10 | resex 6034 | . . . . . . 7 ⊢ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V |
12 | 8, 11 | pm3.2i 469 | . . . . . 6 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V)) |
14 | opvtxfv 28894 | . . . . 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 | eqtrid 2777 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
17 | 16, 7 | eqtrdi 2781 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) |
18 | 2 | fveq2i 6899 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
19 | opiedgfv 28897 | . . . . . 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 | eqtrid 2777 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
22 | eupthvdres.p | . . . . . 6 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
23 | 9 | eupthf1o 30091 | . . . . . 6 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
24 | f1ofo 6845 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
25 | foima 6815 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | |
26 | 22, 23, 24, 25 | 4syl 19 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) |
27 | 26 | reseq2d 5985 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
28 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
29 | 28 | funfnd 6585 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
30 | fnresdm 6675 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼) | |
31 | 29, 30 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼) |
32 | 21, 27, 31 | 3eqtrd 2769 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) |
33 | 32, 9 | eqtrdi 2781 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) |
34 | 1, 5, 17, 33 | vtxdeqd 29368 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 〈cop 4636 class class class wbr 5149 dom cdm 5678 ↾ cres 5680 “ cima 5681 Fun wfun 6543 Fn wfn 6544 –onto→wfo 6547 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 0cc0 11145 ..^cfzo 13667 ♯chash 14330 Vtxcvtx 28886 iEdgciedg 28887 VtxDegcvtxdg 29356 EulerPathsceupth 30084 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-1st 7994 df-2nd 7995 df-vtx 28888 df-iedg 28889 df-vtxdg 29357 df-wlks 29490 df-trls 29583 df-eupth 30085 |
This theorem is referenced by: eupth2 30126 |
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