![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29063) 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 28867 | . . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
2 | upgrres.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | uhgrfun 28829 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
4 | funres 6583 | . . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹)) | |
5 | 1, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → Fun (𝐸 ↾ 𝐹)) |
6 | 5 | funfnd 6572 | . . . 4 ⊢ (𝐺 ∈ UMGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹)) |
7 | 6 | adantr 480 | . . 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 29065 | . . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) |
11 | df-f 6540 | . . 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 582 | . 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 5457 | . . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V | |
15 | 13, 14 | eqeltri 2823 | . . 3 ⊢ 𝑆 ∈ V |
16 | 8, 2, 9, 13 | uhgrspan1lem2 29061 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
17 | 16 | eqcomi 2735 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
18 | 8, 2, 9, 13 | uhgrspan1lem3 29062 | . . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹) |
19 | 18 | eqcomi 2735 | . . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆) |
20 | 17, 19 | isumgrs 28859 | . . 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 257 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∉ wnel 3040 {crab 3426 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 ⟨cop 4629 dom cdm 5669 ran crn 5670 ↾ cres 5671 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 2c2 12268 ♯chash 14292 Vtxcvtx 28759 iEdgciedg 28760 UHGraphcuhgr 28819 UMGraphcumgr 28844 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-hash 14293 df-vtx 28761 df-iedg 28762 df-uhgr 28821 df-upgr 28845 df-umgr 28846 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |