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| 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.) | 
| 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 5469 | . . . 4 ⊢ 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ∈ V | |
| 4 | 2, 3 | eqeltri 2837 | . . 3 ⊢ 𝐻 ∈ V | 
| 5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) | 
| 6 | 2 | fveq2i 6909 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) | 
| 7 | eupthvdres.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 8 | 7 | fvexi 6920 | . . . . . . 7 ⊢ 𝑉 ∈ V | 
| 9 | eupthvdres.i | . . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 10 | 9 | fvexi 6920 | . . . . . . . 8 ⊢ 𝐼 ∈ V | 
| 11 | 10 | resex 6047 | . . . . . . 7 ⊢ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V | 
| 12 | 8, 11 | pm3.2i 470 | . . . . . 6 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) ∈ V) | 
| 13 | 12 | a1i 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) | |
| 15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = 𝑉) | 
| 16 | 6, 15 | eqtrid 2789 | . . 3 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | 
| 17 | 16, 7 | eqtrdi 2793 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺)) | 
| 18 | 2 | fveq2i 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))))) | |
| 20 | 13, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) | 
| 21 | 18, 20 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))) | 
| 22 | eupthvdres.p | . . . . . 6 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
| 23 | 9 | eupthf1o 30223 | . . . . . 6 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) | 
| 24 | f1ofo 6855 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) | |
| 25 | foima 6825 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | |
| 26 | 22, 23, 24, 25 | 4syl 19 | . . . . 5 ⊢ (𝜑 → (𝐹 “ (0..^(♯‘𝐹))) = dom 𝐼) | 
| 27 | 26 | reseq2d 5997 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹)))) = (𝐼 ↾ dom 𝐼)) | 
| 28 | eupthvdres.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
| 29 | 28 | funfnd 6597 | . . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼) | 
| 30 | fnresdm 6687 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (𝐼 ↾ dom 𝐼) = 𝐼) | |
| 31 | 29, 30 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ↾ dom 𝐼) = 𝐼) | 
| 32 | 21, 27, 31 | 3eqtrd 2781 | . . 3 ⊢ (𝜑 → (iEdg‘𝐻) = 𝐼) | 
| 33 | 32, 9 | eqtrdi 2793 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺)) | 
| 34 | 1, 5, 17, 33 | vtxdeqd 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 –onto→wfo 6559 –1-1-onto→wf1o 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 | 
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