Theorem upgrres1 27583

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 27538 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 6737	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of 6700	. . . . 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 6615	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
5		upgrres1.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸)
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝐸)
8		upgrres1.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺)
9	8	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
10		edgupgr 27407	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
11		elpwi 4539	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
12		upgrres1.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (Vtx‘𝐺)
13	11, 12	sseqtrrdi 3968	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ 𝑉)
14	13	3ad2ant1 1131	. . . . . . . . . . . . 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 593	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
17	16	ad4ant13 747	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ⊆ 𝑉)
18		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑁 ∉ 𝑒)
19		elpwdifsn 4719	. . . . . . . . . 10 ⊢ ((𝑒 ∈ 𝐸 ∧ 𝑒 ⊆ 𝑉 ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	7, 17, 18, 19	syl3anc 1369	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
21		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ UPGraph)
22	9	biimpi 215	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
23	10	simp2d 1141	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
24	21, 22, 23	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ≠ ∅)
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ≠ ∅)
26		nelsn 4598	. . . . . . . . . 10 ⊢ (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → ¬ 𝑒 ∈ {∅})
28	20, 27	eldifd 3894	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
29	28	ex 412	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
30	29	ralrimiva 3107	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
31		rabss 4001	. . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
32	30, 31	sylibr 233	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
33	5, 32	eqsstrid 3965	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
34		elrabi 3611	. . . . . . 7 ⊢ (𝑝 ∈ {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} → 𝑝 ∈ 𝐸)
35		edgval 27322	. . . . . . . . . . . 12 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
36	8, 35	eqtri 2766	. . . . . . . . . . 11 ⊢ 𝐸 = ran (iEdg‘𝐺)
37	36	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐸 ↔ 𝑝 ∈ ran (iEdg‘𝐺))
38		eqid 2738	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	12, 38	upgrf 27359	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
40	39	frnd 6592	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
41	40	sseld 3916	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
42	37, 41	syl5bi 241	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (♯‘𝑥) = (♯‘𝑝))
44	43	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑝) ≤ 2))
45	44	elrab 3617	. . . . . . . . . 10 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝑝) ≤ 2))
46	45	simprbi 496	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝑝) ≤ 2)
47	42, 46	syl6 35	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → (♯‘𝑝) ≤ 2))
48	47	adantr 480	. . . . . . 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 2857	. . . . 5 ⊢ (𝑝 ∈ 𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (♯‘𝑝) ≤ 2))
51	50	impcom 407	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑝 ∈ 𝐹) → (♯‘𝑝) ≤ 2)
52	33, 51	ssrabdv 4003	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
53	4, 52	fssd 6602	. 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 5373	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
56	54, 55	eqeltri 2835	. . 3 ⊢ 𝑆 ∈ V
57	12, 8, 5, 54	upgrres1lem2 27581	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
58	57	eqcomi 2747	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
59	12, 8, 5, 54	upgrres1lem3 27582	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
60	59	eqcomi 2747	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
61	58, 60	isupgr 27357	. . 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 256	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048  ∀wral 3063  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ⟨cop 4564   class class class wbr 5070   I cid 5479  dom cdm 5580  ran crn 5581   ↾ cres 5582  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418   ≤ cle 10941  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  UPGraphcupgr 27353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1st 7804  df-2nd 7805  df-vtx 27271  df-iedg 27272  df-edg 27321  df-upgr 27355

This theorem is referenced by:  nbupgrres  27634