Theorem upgrres 29667

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29664) 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 29463	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
2		upgrres.e	. . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺)
3	2	uhgrfun 29427	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
4		funres 6578	. . . . . 6 ⊢ (Fun 𝐸 → Fun (𝐸 ↾ 𝐹))
5	1, 3, 4	3syl 19	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (𝐸 ↾ 𝐹))
6	5	funfnd 6567	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐹) Fn dom (𝐸 ↾ 𝐹))
7	6	adantr 485	. . 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 29665	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
11		df-f 6540	. . 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 594	. 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 5444	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V
15	13, 14	eqeltri 2858	. . 3 ⊢ 𝑆 ∈ V
16	8, 2, 9, 13	uhgrspan1lem2 29662	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
17	16	eqcomi 2771	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
18	8, 2, 9, 13	uhgrspan1lem3 29663	. . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹)
19	18	eqcomi 2771	. . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆)
20	17, 19	isupgr 29445	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
21	15, 20	mp1i 14	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22	12, 21	mpbird 260	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ∉ wnel 3063  {crab 3415  Vcvv 3454   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588  ⟨cop 4594   class class class wbr 5108  dom cdm 5660  ran crn 5661   ↾ cres 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536   ≤ cle 11250  2c2 12301  ♯chash 14373  Vtxcvtx 29357  iEdgciedg 29358  UHGraphcuhgr 29417  UPGraphcupgr 29441

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1st 7984  df-2nd 7985  df-vtx 29359  df-iedg 29360  df-uhgr 29419  df-upgr 29443

This theorem is used by:  finsumvtxdg2size  29911