MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupthvdres Structured version   Visualization version   GIF version

Theorem eupthvdres 30264
Description: Formerly part of proof of eupth2 30268: 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.)
Hypotheses
Ref Expression
eupthvdres.v 𝑉 = (Vtx‘𝐺)
eupthvdres.i 𝐼 = (iEdg‘𝐺)
eupthvdres.g (𝜑𝐺𝑊)
eupthvdres.f (𝜑 → Fun 𝐼)
eupthvdres.p (𝜑𝐹(EulerPaths‘𝐺)𝑃)
eupthvdres.h 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
Assertion
Ref Expression
eupthvdres (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Proof of Theorem eupthvdres
StepHypRef Expression
1 eupthvdres.g . 2 (𝜑𝐺𝑊)
2 eupthvdres.h . . . 4 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩
3 opex 5475 . . . 4 𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ ∈ V
42, 3eqeltri 2835 . . 3 𝐻 ∈ V
54a1i 11 . 2 (𝜑𝐻 ∈ V)
62fveq2i 6910 . . . 4 (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
7 eupthvdres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
87fvexi 6921 . . . . . . 7 𝑉 ∈ V
9 eupthvdres.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
109fvexi 6921 . . . . . . . 8 𝐼 ∈ V
1110resex 6049 . . . . . . 7 (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V
128, 11pm3.2i 470 . . . . . 6 (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V)
1312a1i 11 . . . . 5 (𝜑 → (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V))
14 opvtxfv 29036 . . . . 5 ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = 𝑉)
1513, 14syl 17 . . . 4 (𝜑 → (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = 𝑉)
166, 15eqtrid 2787 . . 3 (𝜑 → (Vtx‘𝐻) = 𝑉)
1716, 7eqtrdi 2791 . 2 (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))
182fveq2i 6910 . . . . 5 (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
19 opiedgfv 29039 . . . . . 6 ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
2013, 19syl 17 . . . . 5 (𝜑 → (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
2118, 20eqtrid 2787 . . . 4 (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
22 eupthvdres.p . . . . . 6 (𝜑𝐹(EulerPaths‘𝐺)𝑃)
239eupthf1o 30233 . . . . . 6 (𝐹(EulerPaths‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
24 f1ofo 6856 . . . . . 6 (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)
25 foima 6826 . . . . . 6 (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼)
2622, 23, 24, 254syl 19 . . . . 5 (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼)
2726reseq2d 6000 . . . 4 (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼))
28 eupthvdres.f . . . . . 6 (𝜑 → Fun 𝐼)
2928funfnd 6599 . . . . 5 (𝜑𝐼 Fn dom 𝐼)
30 fnresdm 6688 . . . . 5 (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼)
3129, 30syl 17 . . . 4 (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼)
3221, 27, 313eqtrd 2779 . . 3 (𝜑 → (iEdg‘𝐻) = 𝐼)
3332, 9eqtrdi 2791 . 2 (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
341, 5, 17, 33vtxdeqd 29510 1 (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  dom cdm 5689  cres 5691  cima 5692  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  0cc0 11153  ..^cfzo 13691  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  VtxDegcvtxdg 29498  EulerPathsceupth 30226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-vtx 29030  df-iedg 29031  df-vtxdg 29499  df-wlks 29632  df-trls 29725  df-eupth 30227
This theorem is referenced by:  eupth2  30268
  Copyright terms: Public domain W3C validator