Theorem upgrres1 26205

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 26160 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 6174	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of 6137	. . . . 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 6063	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
5		upgrres1.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸)
7	6	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝐸)
8		upgrres1.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (Edg‘𝐺)
9	8	eleq2i 2693	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
10		edgupgr 26029	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (#‘𝑒) ≤ 2))
11		elpwi 4168	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
12		upgrres1.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (Vtx‘𝐺)
13	11, 12	syl6sseqr 3652	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ 𝑉)
14	13	3ad2ant1 1082	. . . . . . . . . . . . 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 492	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
17	16	ad4ant13 1292	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ⊆ 𝑉)
18		simpr 477	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑁 ∉ 𝑒)
19		elpwdifsn 4319	. . . . . . . . . 10 ⊢ ((𝑒 ∈ 𝐸 ∧ 𝑒 ⊆ 𝑉 ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
20	7, 17, 18, 19	syl3anc 1326	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
21		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ UPGraph )
22	9	biimpi 206	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
23	10	simp2d 1074	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
24	21, 22, 23	syl2an 494	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → 𝑒 ≠ ∅)
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ≠ ∅)
26		nelsn 4212	. . . . . . . . . 10 ⊢ (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → ¬ 𝑒 ∈ {∅})
28	20, 27	eldifd 3585	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) ∧ 𝑁 ∉ 𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
29	28	ex 450	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
30	29	ralrimiva 2966	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
31		rabss 3679	. . . . . 6 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒 ∈ 𝐸 (𝑁 ∉ 𝑒 → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
32	30, 31	sylibr 224	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
33	5, 32	syl5eqss 3649	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
34		elrabi 3359	. . . . . . 7 ⊢ (𝑝 ∈ {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} → 𝑝 ∈ 𝐸)
35		edgval 25941	. . . . . . . . . . . 12 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
36	8, 35	eqtri 2644	. . . . . . . . . . 11 ⊢ 𝐸 = ran (iEdg‘𝐺)
37	36	eleq2i 2693	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐸 ↔ 𝑝 ∈ ran (iEdg‘𝐺))
38		eqid 2622	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	12, 38	upgrf 25981	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
40		frn 6053	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
42	41	sseld 3602	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
43	37, 42	syl5bi 232	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
44		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (#‘𝑥) = (#‘𝑝))
45	44	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑝) ≤ 2))
46	45	elrab 3363	. . . . . . . . . 10 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝑝) ≤ 2))
47	46	simprbi 480	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (#‘𝑝) ≤ 2)
48	43, 47	syl6 35	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝑝 ∈ 𝐸 → (#‘𝑝) ≤ 2))
49	48	adantr 481	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑝 ∈ 𝐸 → (#‘𝑝) ≤ 2))
50	34, 49	syl5com 31	. . . . . 6 ⊢ (𝑝 ∈ {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (#‘𝑝) ≤ 2))
51	50, 5	eleq2s 2719	. . . . 5 ⊢ (𝑝 ∈ 𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (#‘𝑝) ≤ 2))
52	51	impcom 446	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑝 ∈ 𝐹) → (#‘𝑝) ≤ 2)
53	33, 52	ssrabdv 3681	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
54	4, 53	fssd 6057	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
55		upgrres1.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
56		opex 4932	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
57	55, 56	eqeltri 2697	. . 3 ⊢ 𝑆 ∈ V
58	12, 8, 5, 55	upgrres1lem2 26203	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
59	58	eqcomi 2631	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
60	12, 8, 5, 55	upgrres1lem3 26204	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
61	60	eqcomi 2631	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
62	59, 61	isupgr 25979	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
63	57, 62	mp1i 13	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
64	54, 63	mpbird 247	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ⟨cop 4183   class class class wbr 4653   I cid 5023  dom cdm 5114  ran crn 5115   ↾ cres 5116  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-edg 25940  df-upgr 25977

This theorem is referenced by:  nbupgrres  26266