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Theorem usgrres1 29148
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 29101 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 6882 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of1 6843 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹1-1𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹1-1𝐹)
4 eqidd 2729 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5 dmresi 6060 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
65a1i 11 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7 eqidd 2729 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹 = 𝐹)
84, 6, 7f1eq123d 6836 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ↔ ( I ↾ 𝐹):𝐹1-1𝐹))
93, 8mpbird 256 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹)
10 usgrumgr 29014 . . . 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 29143 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
1510, 14sylan 578 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
16 f1ssr 6805 . . 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 582 . 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 5470 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
2018, 19eqeltri 2825 . . 3 𝑆 ∈ V
2111, 12, 13, 18upgrres1lem2 29144 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
2221eqcomi 2737 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
2311, 12, 13, 18upgrres1lem3 29145 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2423eqcomi 2737 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2522, 24isusgrs 28989 . . 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 256 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wnel 3043  {crab 3430  Vcvv 3473  cdif 3946  wss 3949  𝒫 cpw 4606  {csn 4632  cop 4638   I cid 5579  dom cdm 5682  ran crn 5683  cres 5684  1-1wf1 6550  1-1-ontowf1o 6552  cfv 6553  2c2 12305  chash 14329  Vtxcvtx 28829  iEdgciedg 28830  Edgcedg 28880  UMGraphcumgr 28914  USGraphcusgr 28982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-hash 14330  df-vtx 28831  df-iedg 28832  df-edg 28881  df-uhgr 28891  df-upgr 28915  df-umgr 28916  df-usgr 28984
This theorem is referenced by:  fusgrfis  29163  cusgrres  29282
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