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Theorem usgrres1 26252
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 26205 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 = {𝑒𝐸𝑁𝑒}
upgrres1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
usgrres1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 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.
StepHypRef Expression
1 f1oi 6212 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of1 6174 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹1-1𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹1-1𝐹)
4 eqidd 2652 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5 dmresi 5492 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
65a1i 11 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7 eqidd 2652 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹 = 𝐹)
84, 6, 7f1eq123d 6169 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ↔ ( I ↾ 𝐹):𝐹1-1𝐹))
93, 8mpbird 247 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹)
10 usgrumgr 26119 . . . 4 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
11 upgrres1.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 upgrres1.e . . . . 5 𝐸 = (Edg‘𝐺)
13 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
1411, 12, 13umgrres1lem 26247 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
1510, 14sylan 487 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
16 f1ssr 6145 . . 3 ((( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ∧ ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
179, 15, 16syl2anc 694 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2})
18 upgrres1.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
19 opex 4962 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
2018, 19eqeltri 2726 . . 3 𝑆 ∈ V
2111, 12, 13, 18upgrres1lem2 26248 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
2221eqcomi 2660 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
2311, 12, 13, 18upgrres1lem3 26249 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2423eqcomi 2660 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2522, 24isusgrs 26096 . . 3 (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
2620, 25mp1i 13 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}))
2717, 26mpbird 247 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wnel 2926  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  𝒫 cpw 4191  {csn 4210  cop 4216   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  2c2 11108  #chash 13157  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UMGraphcumgr 26021  USGraphcusgr 26089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-umgr 26023  df-usgr 26091
This theorem is referenced by:  fusgrfis  26267  cusgrres  26400
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