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Mirrors > Home > MPE Home > Th. List > Mathboxes > eupthvdres | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2 41407: 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 4859 | . . . 4 ⊢ 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))〉 ∈ V | |
4 | 2, 3 | eqeltri 2684 | . . 3 ⊢ 𝐻 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
6 | 2 | fveq2i 6106 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))〉) |
7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | fvex 6113 | . . . . . . . 8 ⊢ (Vtx‘𝐺) ∈ V | |
9 | 7, 8 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑉 ∈ V |
10 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
11 | fvex 6113 | . . . . . . . . 9 ⊢ (iEdg‘𝐺) ∈ V | |
12 | 10, 11 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
13 | 12 | resex 5363 | . . . . . . 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 25681 | . . . . 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 2656 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
19 | 18, 7 | syl6eq 2660 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) |
20 | 2 | fveq2i 6106 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))〉) |
21 | opiedgfv 25684 | . . . . . 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 2656 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))) |
24 | eupthvdres.p | . . . . . . 7 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
25 | 10 | eupthf1o 41372 | . . . . . . 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 6057 | . . . . . 6 ⊢ (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(#‘𝐹))–onto→dom 𝐼) | |
28 | foima 6033 | . . . . . 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 5317 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
31 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
32 | funfn 5833 | . . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
33 | 31, 32 | sylib 207 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
34 | fnresdm 5914 | . . . . 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 2648 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) |
37 | 36, 10 | syl6eq 2660 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) |
38 | 1, 5, 19, 37 | vtxdeqd 40692 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 Fn wfn 5799 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 EulerPathsceupth 41364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-vtx 25675 df-iedg 25676 df-vtxdg 40682 df-1wlks 40800 df-trls 40901 df-eupth 41365 |
This theorem is referenced by: eupth2 41407 |
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