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Theorem upgrres 26420
 Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 26417) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.)
Hypotheses
Ref Expression
upgrres.v 𝑉 = (Vtx‘𝐺)
upgrres.e 𝐸 = (iEdg‘𝐺)
upgrres.f 𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
upgrres.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩
Assertion
Ref Expression
upgrres ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
Distinct variable groups:   𝑖,𝐸   𝑖,𝑁
Allowed substitution hints:   𝑆(𝑖)   𝐹(𝑖)   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem upgrres
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 upgruhgr 26217 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2 upgrres.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
32uhgrfun 26181 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4 funres 6072 . . . . . 6 (Fun 𝐸 → Fun (𝐸𝐹))
51, 3, 43syl 18 . . . . 5 (𝐺 ∈ UPGraph → Fun (𝐸𝐹))
65funfnd 6062 . . . 4 (𝐺 ∈ UPGraph → (𝐸𝐹) Fn dom (𝐸𝐹))
76adantr 466 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹) Fn dom (𝐸𝐹))
8 upgrres.v . . . 4 𝑉 = (Vtx‘𝐺)
9 upgrres.f . . . 4 𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
108, 2, 9upgrreslem 26418 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 df-f 6035 . . 3 ((𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸𝐹) Fn dom (𝐸𝐹) ∧ ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
127, 10, 11sylanbrc 564 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
13 upgrres.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩
14 opex 5060 . . . 4 ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩ ∈ V
1513, 14eqeltri 2845 . . 3 𝑆 ∈ V
168, 2, 9, 13uhgrspan1lem2 26415 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
1716eqcomi 2779 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
188, 2, 9, 13uhgrspan1lem3 26416 . . . . 5 (iEdg‘𝑆) = (𝐸𝐹)
1918eqcomi 2779 . . . 4 (𝐸𝐹) = (iEdg‘𝑆)
2017, 19isupgr 26199 . . 3 (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2115, 20mp1i 13 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑆 ∈ UPGraph ↔ (𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
2212, 21mpbird 247 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144   ∉ wnel 3045  {crab 3064  Vcvv 3349   ∖ cdif 3718   ⊆ wss 3721  ∅c0 4061  𝒫 cpw 4295  {csn 4314  ⟨cop 4320   class class class wbr 4784  dom cdm 5249  ran crn 5250   ↾ cres 5251  Fun wfun 6025   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031   ≤ cle 10276  2c2 11271  ♯chash 13320  Vtxcvtx 26094  iEdgciedg 26095  UHGraphcuhgr 26171  UPGraphcupgr 26195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-1st 7314  df-2nd 7315  df-vtx 26096  df-iedg 26097  df-uhgr 26173  df-upgr 26197 This theorem is referenced by:  finsumvtxdg2size  26680
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