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Mirrors > Home > MPE Home > Th. List > eupthvdres | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2 28018: 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 5356 | . . . 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 6673 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 7 | fvexi 6684 | . . . . . . 7 ⊢ 𝑉 ∈ V |
9 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
10 | 9 | fvexi 6684 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
11 | 10 | resex 5899 | . . . . . . 7 ⊢ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V |
12 | 8, 11 | pm3.2i 473 | . . . . . 6 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V)) |
14 | opvtxfv 26789 | . . . . 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 6673 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
19 | opiedgfv 26792 | . . . . . 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 27983 | . . . . . . 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 6622 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
26 | foima 6595 | . . . . . 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 5853 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
29 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
30 | 29 | funfnd 6386 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
31 | fnresdm 6466 | . . . . 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 27259 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 dom cdm 5555 ↾ cres 5557 “ cima 5558 Fun wfun 6349 Fn wfn 6350 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ..^cfzo 13034 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 VtxDegcvtxdg 27247 EulerPathsceupth 27976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-vtx 26783 df-iedg 26784 df-vtxdg 27248 df-wlks 27381 df-trls 27474 df-eupth 27977 |
This theorem is referenced by: eupth2 28018 |
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