Theorem upgrres 29392

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29389) 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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
upgrres.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩

Assertion

Ref	Expression
upgrres	⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 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.

Step	Hyp	Ref	Expression
1		upgruhgr 29188	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2		upgrres.e	. . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺)
3	2	uhgrfun 29152	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
4		funres 6535	. . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹))
5	1, 3, 4	3syl 18	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (𝐸 ↾ 𝐹))
6	5	funfnd 6524	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹))
7	6	adantr 480	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹))
8		upgrres.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
9		upgrres.f	. . . 4 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
10	8, 2, 9	upgrreslem 29390	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11		df-f 6497	. . 3 ⊢ ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ((𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹) ∧ ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
12	7, 10, 11	sylanbrc 584	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
13		upgrres.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩
14		opex 5412	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V
15	13, 14	eqeltri 2833	. . 3 ⊢ 𝑆 ∈ V
16	8, 2, 9, 13	uhgrspan1lem2 29387	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
17	16	eqcomi 2746	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
18	8, 2, 9, 13	uhgrspan1lem3 29388	. . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹)
19	18	eqcomi 2746	. . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆)
20	17, 19	isupgr 29170	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
21	15, 20	mp1i 13	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22	12, 21	mpbird 257	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ⟨cop 4574   class class class wbr 5086  dom cdm 5625  ran crn 5626   ↾ cres 5627  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493   ≤ cle 11174  2c2 12230  ♯chash 14286  Vtxcvtx 29082  iEdgciedg 29083  UHGraphcuhgr 29142  UPGraphcupgr 29166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-1st 7936  df-2nd 7937  df-vtx 29084  df-iedg 29085  df-uhgr 29144  df-upgr 29168

This theorem is referenced by:  finsumvtxdg2size  29637