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Mirrors > Home > MPE Home > Th. List > umgrres | Structured version Visualization version GIF version |
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 29136) is a multigraph. (Contributed by AV, 27-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 |
---|---|
umgrres | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgruhgr 28937 | . . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
2 | upgrres.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 28899 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | funres 6600 | . . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹)) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → Fun (𝐸 ↾ 𝐹)) |
6 | 5 | funfnd 6589 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) |
7 | 6 | adantr 479 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 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 | umgrreslem 29138 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
11 | df-f 6557 | . . 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 581 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
13 | upgrres.s | . . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ | |
14 | opex 5470 | . . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V | |
15 | 13, 14 | eqeltri 2825 | . . 3 ⊢ 𝑆 ∈ V |
16 | 8, 2, 9, 13 | uhgrspan1lem2 29134 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
17 | 16 | eqcomi 2737 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
18 | 8, 2, 9, 13 | uhgrspan1lem3 29135 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) |
19 | 18 | eqcomi 2737 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) |
20 | 17, 19 | isumgrs 28929 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UMGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
21 | 15, 20 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UMGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})) |
22 | 12, 21 | mpbird 256 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∉ wnel 3043 {crab 3430 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 𝒫 cpw 4606 {csn 4632 ⟨cop 4638 dom cdm 5682 ran crn 5683 ↾ cres 5684 Fun wfun 6547 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 2c2 12305 ♯chash 14329 Vtxcvtx 28829 iEdgciedg 28830 UHGraphcuhgr 28889 UMGraphcumgr 28914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 df-vtx 28831 df-iedg 28832 df-uhgr 28891 df-upgr 28915 df-umgr 28916 |
This theorem is referenced by: (None) |
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