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Theorem upgrres 29139
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29136) 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 28935 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2 upgrres.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
32uhgrfun 28899 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4 funres 6600 . . . . . 6 (Fun 𝐸 → Fun (𝐸𝐹))
51, 3, 43syl 18 . . . . 5 (𝐺 ∈ UPGraph → Fun (𝐸𝐹))
65funfnd 6589 . . . 4 (𝐺 ∈ UPGraph → (𝐸𝐹) Fn dom (𝐸𝐹))
76adantr 479 . . 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 29137 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 df-f 6557 . . 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 581 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
13 upgrres.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩
14 opex 5470 . . . 4 ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩ ∈ V
1513, 14eqeltri 2825 . . 3 𝑆 ∈ V
168, 2, 9, 13uhgrspan1lem2 29134 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
1716eqcomi 2737 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
188, 2, 9, 13uhgrspan1lem3 29135 . . . . 5 (iEdg‘𝑆) = (𝐸𝐹)
1918eqcomi 2737 . . . 4 (𝐸𝐹) = (iEdg‘𝑆)
2017, 19isupgr 28917 . . 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 256 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
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  c0 4326  𝒫 cpw 4606  {csn 4632  cop 4638   class class class wbr 5152  dom cdm 5682  ran crn 5683  cres 5684  Fun wfun 6547   Fn wfn 6548  wf 6549  cfv 6553  cle 11287  2c2 12305  chash 14329  Vtxcvtx 28829  iEdgciedg 28830  UHGraphcuhgr 28889  UPGraphcupgr 28913
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-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  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-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-1st 7999  df-2nd 8000  df-vtx 28831  df-iedg 28832  df-uhgr 28891  df-upgr 28915
This theorem is referenced by:  finsumvtxdg2size  29384
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