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Theorem upgrres 29245
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29242) 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 29041 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2 upgrres.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
32uhgrfun 29005 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4 funres 6603 . . . . . 6 (Fun 𝐸 → Fun (𝐸𝐹))
51, 3, 43syl 18 . . . . 5 (𝐺 ∈ UPGraph → Fun (𝐸𝐹))
65funfnd 6592 . . . 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 29243 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 df-f 6560 . . 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 5472 . . . 4 ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩ ∈ V
1513, 14eqeltri 2822 . . 3 𝑆 ∈ V
168, 2, 9, 13uhgrspan1lem2 29240 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
1716eqcomi 2735 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
188, 2, 9, 13uhgrspan1lem3 29241 . . . . 5 (iEdg‘𝑆) = (𝐸𝐹)
1918eqcomi 2735 . . . 4 (𝐸𝐹) = (iEdg‘𝑆)
2017, 19isupgr 29023 . . 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 1534  wcel 2099  wnel 3036  {crab 3419  Vcvv 3462  cdif 3944  wss 3947  c0 4325  𝒫 cpw 4607  {csn 4633  cop 4639   class class class wbr 5155  dom cdm 5684  ran crn 5685  cres 5686  Fun wfun 6550   Fn wfn 6551  wf 6552  cfv 6556  cle 11301  2c2 12321  chash 14349  Vtxcvtx 28935  iEdgciedg 28936  UHGraphcuhgr 28995  UPGraphcupgr 29019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fv 6564  df-1st 8005  df-2nd 8006  df-vtx 28937  df-iedg 28938  df-uhgr 28997  df-upgr 29021
This theorem is referenced by:  finsumvtxdg2size  29490
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