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Theorem usgrres1 29075
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 29028 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 6864 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of1 6825 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹1-1𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹1-1𝐹)
4 eqidd 2727 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5 dmresi 6044 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
65a1i 11 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7 eqidd 2727 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹 = 𝐹)
84, 6, 7f1eq123d 6818 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ↔ ( I ↾ 𝐹):𝐹1-1𝐹))
93, 8mpbird 257 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹)
10 usgrumgr 28942 . . . 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 29070 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
1510, 14sylan 579 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
16 f1ssr 6787 . . 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 583 . 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 5457 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
2018, 19eqeltri 2823 . . 3 𝑆 ∈ V
2111, 12, 13, 18upgrres1lem2 29071 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
2221eqcomi 2735 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
2311, 12, 13, 18upgrres1lem3 29072 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2423eqcomi 2735 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2522, 24isusgrs 28919 . . 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 257 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wnel 3040  {crab 3426  Vcvv 3468  cdif 3940  wss 3943  𝒫 cpw 4597  {csn 4623  cop 4629   I cid 5566  dom cdm 5669  ran crn 5670  cres 5671  1-1wf1 6533  1-1-ontowf1o 6535  cfv 6536  2c2 12268  chash 14292  Vtxcvtx 28759  iEdgciedg 28760  Edgcedg 28810  UMGraphcumgr 28844  USGraphcusgr 28912
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-hash 14293  df-vtx 28761  df-iedg 28762  df-edg 28811  df-uhgr 28821  df-upgr 28845  df-umgr 28846  df-usgr 28914
This theorem is referenced by:  fusgrfis  29090  cusgrres  29209
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