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Theorem cusgrres 26325
Description: Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
Hypotheses
Ref Expression
cusgrres.v 𝑉 = (Vtx‘𝐺)
cusgrres.e 𝐸 = (Edg‘𝐺)
cusgrres.f 𝐹 = {𝑒𝐸𝑁𝑒}
cusgrres.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
cusgrres ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑆 ∈ ComplUSGraph)
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem cusgrres
Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusgrusgr 26296 . . 3 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )
2 cusgrres.v . . . 4 𝑉 = (Vtx‘𝐺)
3 cusgrres.e . . . 4 𝐸 = (Edg‘𝐺)
4 cusgrres.f . . . 4 𝐹 = {𝑒𝐸𝑁𝑒}
5 cusgrres.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
62, 3, 4, 5usgrres1 26188 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph )
71, 6sylan 488 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph )
8 iscusgr 26295 . . . 4 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
9 usgrupgr 26058 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
109adantr 481 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ UPGraph )
1110anim1i 591 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁𝑉) → (𝐺 ∈ UPGraph ∧ 𝑁𝑉))
1211anim1i 591 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})))
132iscplgr 26291 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑛𝑉 𝑛 ∈ (UnivVtx‘𝐺)))
14 eldifi 3724 . . . . . . . . . . . . 13 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑉)
1514ad2antll 764 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ (𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁}))) → 𝑣𝑉)
16 eleq1 2687 . . . . . . . . . . . . 13 (𝑛 = 𝑣 → (𝑛 ∈ (UnivVtx‘𝐺) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
1716rspcv 3300 . . . . . . . . . . . 12 (𝑣𝑉 → (∀𝑛𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
1815, 17syl 17 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ (𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁}))) → (∀𝑛𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
1918ex 450 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁})) → (∀𝑛𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺))))
2019com23 86 . . . . . . . . 9 (𝐺 ∈ USGraph → (∀𝑛𝑉 𝑛 ∈ (UnivVtx‘𝐺) → ((𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
2113, 20sylbid 230 . . . . . . . 8 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph → ((𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
2221imp 445 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → ((𝑁𝑉𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)))
2322impl 649 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))
242, 3, 4, 5uvtxupgrres 26290 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝑆)))
2512, 23, 24sylc 65 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝑆))
2625ralrimiva 2963 . . . 4 (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
278, 26sylanb 489 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
28 opex 4923 . . . . 5 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
295, 28eqeltri 2695 . . . 4 𝑆 ∈ V
302, 3, 4, 5upgrres1lem2 26184 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
3130eqcomi 2629 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
3231iscplgr 26291 . . . 4 (𝑆 ∈ V → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
3329, 32mp1i 13 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
3427, 33mpbird 247 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑆 ∈ ComplGraph)
35 iscusgr 26295 . 2 (𝑆 ∈ ComplUSGraph ↔ (𝑆 ∈ USGraph ∧ 𝑆 ∈ ComplGraph))
367, 34, 35sylanbrc 697 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑆 ∈ ComplUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wnel 2894  wral 2909  {crab 2913  Vcvv 3195  cdif 3564  {csn 4168  cop 4174   I cid 5013  cres 5106  cfv 5876  Vtxcvtx 25855  Edgcedg 25920   UPGraph cupgr 25956   USGraph cusgr 26025  UnivVtxcuvtxa 26206  ComplGraphccplgr 26207  ComplUSGraphccusgr 26208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-vtx 25857  df-iedg 25858  df-edg 25921  df-uhgr 25934  df-upgr 25958  df-umgr 25959  df-uspgr 26026  df-usgr 26027  df-nbgr 26209  df-uvtxa 26211  df-cplgr 26212  df-cusgr 26213
This theorem is referenced by:  cusgrsize  26331
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