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| Mirrors > Home > MPE Home > Th. List > upgrres | Structured version Visualization version GIF version | ||
| Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29320) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| upgrres.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| upgrres.e | ⊢ 𝐸 = (iEdg‘𝐺) | 
| upgrres.f | ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | 
| upgrres.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 | 
| Ref | Expression | 
|---|---|
| upgrres | ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | upgruhgr 29119 | . . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
| 2 | upgrres.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 3 | 2 | uhgrfun 29083 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) | 
| 4 | funres 6608 | . . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹)) | |
| 5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (𝐸 ↾ 𝐹)) | 
| 6 | 5 | funfnd 6597 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) | 
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) | 
| 8 | upgrres.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 9 | upgrres.f | . . . 4 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
| 10 | 8, 2, 9 | upgrreslem 29321 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) | 
| 11 | df-f 6565 | . . 3 ⊢ ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹) ∧ ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) | |
| 12 | 7, 10, 11 | sylanbrc 583 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) | 
| 13 | upgrres.s | . . . 4 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 | |
| 14 | opex 5469 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ∈ V | |
| 15 | 13, 14 | eqeltri 2837 | . . 3 ⊢ 𝑆 ∈ V | 
| 16 | 8, 2, 9, 13 | uhgrspan1lem2 29318 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) | 
| 17 | 16 | eqcomi 2746 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) | 
| 18 | 8, 2, 9, 13 | uhgrspan1lem3 29319 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) | 
| 19 | 18 | eqcomi 2746 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) | 
| 20 | 17, 19 | isupgr 29101 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) | 
| 21 | 15, 20 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})) | 
| 22 | 12, 21 | mpbird 257 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 ran crn 5686 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ≤ cle 11296 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 UPGraphcupgr 29097 | 
| 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-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-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-fv 6569 df-1st 8014 df-2nd 8015 df-vtx 29015 df-iedg 29016 df-uhgr 29075 df-upgr 29099 | 
| This theorem is referenced by: finsumvtxdg2size 29568 | 
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