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Theorem umgrres1 26094
 Description: A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 26053 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.)
Hypotheses
Ref Expression
upgrres1.v 𝑉 = (Vtx‘𝐺)
upgrres1.e 𝐸 = (Edg‘𝐺)
upgrres1.f 𝐹 = {𝑒𝐸𝑁𝑒}
upgrres1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
umgrres1 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph )
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem umgrres1
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 f1oi 6131 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of 6094 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹𝐹)
4 dmresi 5416 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
54a1i 11 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
65feq2d 5988 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹𝐹))
73, 6mpbird 247 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
8 rnresi 5438 . . . 4 ran ( I ↾ 𝐹) = 𝐹
9 upgrres1.v . . . . 5 𝑉 = (Vtx‘𝐺)
10 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
11 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
129, 10, 11umgrres1lem 26090 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
138, 12syl5eqssr 3629 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝐹 ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
147, 13fssd 6014 . 2 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
15 upgrres1.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
16 opex 4893 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
1715, 16eqeltri 2694 . . 3 𝑆 ∈ V
189, 10, 11, 15upgrres1lem2 26091 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
1918eqcomi 2630 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
209, 10, 11, 15upgrres1lem3 26092 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2120eqcomi 2630 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2219, 21isumgrs 25886 . . 3 (𝑆 ∈ V → (𝑆 ∈ UMGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
2317, 22mp1i 13 . 2 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → (𝑆 ∈ UMGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
2414, 23mpbird 247 1 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ∉ wnel 2893  {crab 2911  Vcvv 3186   ∖ cdif 3552  𝒫 cpw 4130  {csn 4148  ⟨cop 4154   I cid 4984  dom cdm 5074  ran crn 5075   ↾ cres 5076  ⟶wf 5843  –1-1-onto→wf1o 5846  ‘cfv 5847  2c2 11014  #chash 13057  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   UMGraph cumgr 25872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-vtx 25776  df-iedg 25777  df-edg 25840  df-uhgr 25849  df-upgr 25873  df-umgr 25874 This theorem is referenced by: (None)
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