Theorem usgrres1 27585

Description: Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted 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 Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-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
usgrres1	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

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

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

Proof of Theorem usgrres1

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6737	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of1 6699	. . . . 5 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹–1-1→𝐹)
3	1, 2	mp1i 13	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):𝐹–1-1→𝐹)
4		eqidd 2739	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5		dmresi 5950	. . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7		eqidd 2739	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = 𝐹)
8	4, 6, 7	f1eq123d 6692	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ↔ ( I ↾ 𝐹):𝐹–1-1→𝐹))
9	3, 8	mpbird 256	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹)
10		usgrumgr 27452	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
11		upgrres1.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
13		upgrres1.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
14	11, 12, 13	umgrres1lem 27580	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
15	10, 14	sylan 579	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
16		f1ssr 6661	. . 3 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ∧ ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
17	9, 15, 16	syl2anc 583	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
18		upgrres1.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
19		opex 5373	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
20	18, 19	eqeltri 2835	. . 3 ⊢ 𝑆 ∈ V
21	11, 12, 13, 18	upgrres1lem2 27581	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
22	21	eqcomi 2747	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
23	11, 12, 13, 18	upgrres1lem3 27582	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
24	23	eqcomi 2747	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
25	22, 24	isusgrs 27429	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
26	20, 25	mp1i 13	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
27	17, 26	mpbird 256	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∉ wnel 3048  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ⟨cop 4564   I cid 5479  dom cdm 5580  ran crn 5581   ↾ cres 5582  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  UMGraphcumgr 27354  USGraphcusgr 27422

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  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  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-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  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-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  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-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-vtx 27271  df-iedg 27272  df-edg 27321  df-uhgr 27331  df-upgr 27355  df-umgr 27356  df-usgr 27424

This theorem is referenced by:  fusgrfis  27600  cusgrres  27718