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

Theorem eupthvdres 30254
Description: Formerly part of proof of eupth2 30258: 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 5469 . . . 4 𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩ ∈ V
42, 3eqeltri 2837 . . 3 𝐻 ∈ V
54a1i 11 . 2 (𝜑𝐻 ∈ V)
62fveq2i 6909 . . . 4 (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
7 eupthvdres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
87fvexi 6920 . . . . . . 7 𝑉 ∈ V
9 eupthvdres.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
109fvexi 6920 . . . . . . . 8 𝐼 ∈ V
1110resex 6047 . . . . . . 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 29021 . . . . 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 2789 . . 3 (𝜑 → (Vtx‘𝐻) = 𝑉)
1716, 7eqtrdi 2793 . 2 (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))
182fveq2i 6909 . . . . 5 (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩)
19 opiedgfv 29024 . . . . . 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 2789 . . . 4 (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))))
22 eupthvdres.p . . . . . 6 (𝜑𝐹(EulerPaths‘𝐺)𝑃)
239eupthf1o 30223 . . . . . 6 (𝐹(EulerPaths‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
24 f1ofo 6855 . . . . . 6 (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)
25 foima 6825 . . . . . 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 5997 . . . 4 (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼))
28 eupthvdres.f . . . . . 6 (𝜑 → Fun 𝐼)
2928funfnd 6597 . . . . 5 (𝜑𝐼 Fn dom 𝐼)
30 fnresdm 6687 . . . . 5 (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼)
3129, 30syl 17 . . . 4 (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼)
3221, 27, 313eqtrd 2781 . . 3 (𝜑 → (iEdg‘𝐻) = 𝐼)
3332, 9eqtrdi 2793 . 2 (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
341, 5, 17, 33vtxdeqd 29495 1 (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  dom cdm 5685  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  ..^cfzo 13694  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  VtxDegcvtxdg 29483  EulerPathsceupth 30216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-vtxdg 29484  df-wlks 29617  df-trls 29710  df-eupth 30217
This theorem is referenced by:  eupth2  30258
  Copyright terms: Public domain W3C validator