Theorem usgrres 29325

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 29320) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (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
usgrres	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Distinct variable groups:   𝑖,𝐸   𝑖,𝑁

Allowed substitution hints:   𝑆(𝑖)   𝐹(𝑖)   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem usgrres

Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrres.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgrres.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgrf 29172	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		upgrres.f	. . . . . . 7 ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
5	4	ssrab3 4082	. . . . . 6 ⊢ 𝐹 ⊆ dom 𝐸
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ dom 𝐸)
7		f1ssres 6811	. . . . 5 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ 𝐹 ⊆ dom 𝐸) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
8	3, 6, 7	syl2an2r 685	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
9		usgrumgr 29198	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
10	1, 2, 4	umgrreslem 29322	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
11	9, 10	sylan 580	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
12		f1ssr 6810	. . . 4 ⊢ (((𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
13	8, 11, 12	syl2anc 584	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
14		ssdmres 6031	. . . . 5 ⊢ (𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐹) = 𝐹)
15	5, 14	mpbi 230	. . . 4 ⊢ dom (𝐸 ↾ 𝐹) = 𝐹
16		f1eq2 6800	. . . 4 ⊢ (dom (𝐸 ↾ 𝐹) = 𝐹 → ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2} ↔ (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
17	15, 16	ax-mp 5	. . 3 ⊢ ((𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2} ↔ (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
18	13, 17	sylibr 234	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
19		upgrres.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩
20		opex 5469	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V
21	19, 20	eqeltri 2837	. . 3 ⊢ 𝑆 ∈ V
22	1, 2, 4, 19	uhgrspan1lem2 29318	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
23	22	eqcomi 2746	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
24	1, 2, 4, 19	uhgrspan1lem3 29319	. . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹)
25	24	eqcomi 2746	. . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆)
26	23, 25	isusgrs 29173	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
27	21, 26	mp1i 13	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ USGraph ↔ (𝐸 ↾ 𝐹):dom (𝐸 ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
28	18, 27	mpbird 257	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∉ wnel 3046  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ⟨cop 4632  dom cdm 5685  ran crn 5686   ↾ cres 5687  –1-1→wf1 6558  ‘cfv 6561  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  UMGraphcumgr 29098  USGraphcusgr 29166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-vtx 29015  df-iedg 29016  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-usgr 29168

This theorem is referenced by: (None)