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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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