Theorem upgrres1 27355

Description: A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 27310 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	⊢ 𝑉 = (Vtx‘𝐺)
upgrres1.e	⊢ 𝐸 = (Edg‘𝐺)
upgrres1.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
upgrres1.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
upgrres1	⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem upgrres1

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

Step	Hyp	Ref	Expression
1		f1oi 6676	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of 6639	. . . . 5 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹⟶𝐹)
3	1, 2	mp1i 13	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):𝐹⟶𝐹)
4	3	ffdmd 6554	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
5		upgrres1.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		simpr 488	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸)
7	6	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝐸)
8		upgrres1.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺)
9	8	eleq2i 2822	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
10		edgupgr 27179	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
11		elpwi 4508	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
12		upgrres1.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (Vtx‘𝐺)
13	11, 12	sseqtrrdi 3938	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ 𝑉)
14	13	3ad2ant1 1135	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2) → 𝑒 ⊆ 𝑉)
15	10, 14	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ⊆ 𝑉)
16	9, 15	sylan2b 597	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
17	16	ad4ant13 751	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ⊆ 𝑉)
18		simpr 488	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑁 ∉ 𝑒)
19		elpwdifsn 4688	. . . . . . . . . 10 ⊢ ((𝑒 ∈ 𝐸 ∧ 𝑒 ⊆ 𝑉 ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	7, 17, 18, 19	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
21		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ UPGraph)
22	9	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
23	10	simp2d 1145	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
24	21, 22, 23	syl2an 599	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ≠ ∅)
25	24	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ≠ ∅)
26		nelsn 4567	. . . . . . . . . 10 ⊢ (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → ¬ 𝑒 ∈ {∅})
28	20, 27	eldifd 3864	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
29	28	ex 416	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
30	29	ralrimiva 3095	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
31		rabss 3971	. . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
32	30, 31	sylibr 237	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
33	5, 32	eqsstrid 3935	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
34		elrabi 3585	. . . . . . 7 ⊢ (𝑝 ∈ {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} → 𝑝 ∈ 𝐸)
35		edgval 27094	. . . . . . . . . . . 12 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
36	8, 35	eqtri 2759	. . . . . . . . . . 11 ⊢ 𝐸 = ran (iEdg‘𝐺)
37	36	eleq2i 2822	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐸 ↔ 𝑝 ∈ ran (iEdg‘𝐺))
38		eqid 2736	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	12, 38	upgrf 27131	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
40	39	frnd 6531	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
41	40	sseld 3886	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
42	37, 41	syl5bi 245	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (♯‘𝑥) = (♯‘𝑝))
44	43	breq1d 5049	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑝) ≤ 2))
45	44	elrab 3591	. . . . . . . . . 10 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝑝) ≤ 2))
46	45	simprbi 500	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝑝) ≤ 2)
47	42, 46	syl6 35	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → (♯‘𝑝) ≤ 2))
48	47	adantr 484	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑝 ∈ 𝐸 → (♯‘𝑝) ≤ 2))
49	34, 48	syl5com 31	. . . . . 6 ⊢ (𝑝 ∈ {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (♯‘𝑝) ≤ 2))
50	49, 5	eleq2s 2849	. . . . 5 ⊢ (𝑝 ∈ 𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (♯‘𝑝) ≤ 2))
51	50	impcom 411	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑝 ∈ 𝐹) → (♯‘𝑝) ≤ 2)
52	33, 51	ssrabdv 3973	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
53	4, 52	fssd 6541	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
54		upgrres1.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
55		opex 5333	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
56	54, 55	eqeltri 2827	. . 3 ⊢ 𝑆 ∈ V
57	12, 8, 5, 54	upgrres1lem2 27353	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
58	57	eqcomi 2745	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
59	12, 8, 5, 54	upgrres1lem3 27354	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
60	59	eqcomi 2745	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
61	58, 60	isupgr 27129	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
62	56, 61	mp1i 13	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
63	53, 62	mpbird 260	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  {crab 3055  Vcvv 3398   ∖ cdif 3850   ⊆ wss 3853  ∅c0 4223  𝒫 cpw 4499  {csn 4527  ⟨cop 4533   class class class wbr 5039   I cid 5439  dom cdm 5536  ran crn 5537   ↾ cres 5538  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358   ≤ cle 10833  2c2 11850  ♯chash 13861  Vtxcvtx 27041  iEdgciedg 27042  Edgcedg 27092  UPGraphcupgr 27125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-1st 7739  df-2nd 7740  df-vtx 27043  df-iedg 27044  df-edg 27093  df-upgr 27127

This theorem is referenced by:  nbupgrres  27406