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Theorem upgrres 26397
Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 26394) 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 26196 . . . . . 6 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2 upgrres.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
32uhgrfun 26160 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐸)
4 funres 6090 . . . . . 6 (Fun 𝐸 → Fun (𝐸𝐹))
51, 3, 43syl 18 . . . . 5 (𝐺 ∈ UPGraph → Fun (𝐸𝐹))
65funfnd 6080 . . . 4 (𝐺 ∈ UPGraph → (𝐸𝐹) Fn dom (𝐸𝐹))
76adantr 472 . . 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 26395 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11 df-f 6053 . . 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 701 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐸𝐹):dom (𝐸𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
13 upgrres.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩
14 opex 5081 . . . 4 ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩ ∈ V
1513, 14eqeltri 2835 . . 3 𝑆 ∈ V
168, 2, 9, 13uhgrspan1lem2 26392 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
1716eqcomi 2769 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
188, 2, 9, 13uhgrspan1lem3 26393 . . . . 5 (iEdg‘𝑆) = (𝐸𝐹)
1918eqcomi 2769 . . . 4 (𝐸𝐹) = (iEdg‘𝑆)
2017, 19isupgr 26178 . . 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 383   = wceq 1632  wcel 2139  wnel 3035  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  cop 4327   class class class wbr 4804  dom cdm 5266  ran crn 5267  cres 5268  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  cle 10267  2c2 11262  chash 13311  Vtxcvtx 26073  iEdgciedg 26074  UHGraphcuhgr 26150  UPGraphcupgr 26174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-1st 7333  df-2nd 7334  df-vtx 26075  df-iedg 26076  df-uhgr 26152  df-upgr 26176
This theorem is referenced by:  finsumvtxdg2size  26656
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