Theorem usgrres 26200

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 26195) 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 26050	. . . . 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 3688	. . . . . 6 ⊢ 𝐹 ⊆ dom 𝐸
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ dom 𝐸)
7		f1ssres 6108	. . . . 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 876	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
9		usgrumgr 26074	. . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
10	1, 2, 4	umgrreslem 26197	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
11	9, 10	sylan 488	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
12		f1ssr 6107	. . . 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 693	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ 𝐹):𝐹–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
14		ssdmres 5420	. . . . 5 ⊢ (𝐹 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐹) = 𝐹)
15	5, 14	mpbi 220	. . . 4 ⊢ dom (𝐸 ↾ 𝐹) = 𝐹
16		f1eq2 6097	. . . 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 224	. 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 4932	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ∈ V
21	19, 20	eqeltri 2697	. . 3 ⊢ 𝑆 ∈ V
22	1, 2, 4, 19	uhgrspan1lem2 26193	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
23	22	eqcomi 2631	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
24	1, 2, 4, 19	uhgrspan1lem3 26194	. . . . 5 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐹)
25	24	eqcomi 2631	. . . 4 ⊢ (𝐸 ↾ 𝐹) = (iEdg‘𝑆)
26	23, 25	isusgrs 26051	. . 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 247	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph )

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ⟨cop 4183  dom cdm 5114  ran crn 5115   ↾ cres 5116  –1-1→wf1 5885  ‘cfv 5888  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UMGraph cumgr 25976   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-vtx 25876  df-iedg 25877  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-usgr 26046

This theorem is referenced by: (None)