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| Mirrors > Home > ILE Home > Th. List > eupthvdres | GIF version | ||
| Description: 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 | . . 3 ⊢ 𝐻 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 | |
| 3 | eupthvdres.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | vtxex 16242 | . . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V) | |
| 5 | 1, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) ∈ V) |
| 6 | 3, 5 | eqeltrid 2325 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) |
| 7 | eupthvdres.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 8 | iedgex 16243 | . . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (iEdg‘𝐺) ∈ V) | |
| 9 | 1, 8 | syl 14 | . . . . . 6 ⊢ (𝜑 → (iEdg‘𝐺) ∈ V) |
| 10 | 7, 9 | eqeltrid 2325 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
| 11 | resexg 5101 | . . . . 5 ⊢ (𝐼 ∈ V → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) | |
| 12 | 10, 11 | syl 14 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) |
| 13 | opexg 4366 | . . . 4 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V) | |
| 14 | 6, 12, 13 | syl2anc 415 | . . 3 ⊢ (𝜑 → 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V) |
| 15 | 2, 14 | eqeltrid 2325 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
| 16 | 2 | fveq2i 5696 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
| 17 | opvtxfv 16246 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) | |
| 18 | 6, 12, 17 | syl2anc 415 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) |
| 19 | 16, 18 | eqtrid 2283 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
| 20 | 19, 3 | eqtrdi 2287 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) |
| 21 | 2 | fveq2i 5696 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) |
| 22 | opiedgfv 16249 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) | |
| 23 | 6, 12, 22 | syl2anc 415 | . . . . 5 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
| 24 | 21, 23 | eqtrid 2283 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) |
| 25 | eupthvdres.p | . . . . . 6 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
| 26 | 7 | eupthf1o 16674 | . . . . . 6 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) |
| 27 | f1ofo 5644 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
| 28 | foima 5618 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | |
| 29 | 25, 26, 27, 28 | 4syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) |
| 30 | 29 | reseq2d 5061 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) |
| 31 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
| 32 | 31 | funfnd 5406 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) |
| 33 | fnresdm 5490 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼) | |
| 34 | 32, 33 | syl 14 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼) |
| 35 | 24, 30, 34 | 3eqtrd 2275 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) |
| 36 | 35, 7 | eqtrdi 2287 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) |
| 37 | 1, 15, 20, 36 | vtxdeqd 16520 | 1 ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1402 ∈ wcel 2209 Vcvv 2821 〈cop 3711 class class class wbr 4128 dom cdm 4772 ↾ cres 4774 “ cima 4775 Fun wfun 5369 Fn wfn 5370 –onto→wfo 5373 –1-1-onto→wf1o 5374 ‘cfv 5375 (class class class)co 6079 0cc0 8173 ..^cfzo 10532 ♯chash 11197 Vtxcvtx 16236 iEdgciedg 16237 VtxDegcvtxdg 16510 EulerPathsceupth 16666 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 |
| This theorem depends on definitions: df-bi 117 df-dc 847 df-ifp 991 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-1o 6681 df-er 6801 df-map 6918 df-en 7017 df-dom 7018 df-fin 7019 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-z 9628 df-dec 9761 df-uz 9905 df-fz 10395 df-fzo 10533 df-ihash 11198 df-word 11288 df-ndx 13338 df-slot 13339 df-base 13341 df-edgf 16229 df-vtx 16238 df-iedg 16239 df-vtxdg 16511 df-wlks 16542 df-trls 16605 df-eupth 16667 |
| This theorem is referenced by: eupth2fi 16703 |
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