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Theorem usgrres1 27204
 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 27157 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 6639 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of1 6601 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹1-1𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹1-1𝐹)
4 eqidd 2759 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5 dmresi 5893 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
65a1i 11 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7 eqidd 2759 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹 = 𝐹)
84, 6, 7f1eq123d 6594 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ↔ ( I ↾ 𝐹):𝐹1-1𝐹))
93, 8mpbird 260 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹)
10 usgrumgr 27071 . . . 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 27199 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
1510, 14sylan 583 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
16 f1ssr 6567 . . 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 587 . 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 5324 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
2018, 19eqeltri 2848 . . 3 𝑆 ∈ V
2111, 12, 13, 18upgrres1lem2 27200 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
2221eqcomi 2767 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
2311, 12, 13, 18upgrres1lem3 27201 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2423eqcomi 2767 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2522, 24isusgrs 27048 . . 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 260 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∉ wnel 3055  {crab 3074  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  𝒫 cpw 4494  {csn 4522  ⟨cop 4528   I cid 5429  dom cdm 5524  ran crn 5525   ↾ cres 5526  –1-1→wf1 6332  –1-1-onto→wf1o 6334  ‘cfv 6335  2c2 11729  ♯chash 13740  Vtxcvtx 26888  iEdgciedg 26889  Edgcedg 26939  UMGraphcumgr 26973  USGraphcusgr 27041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-hash 13741  df-vtx 26890  df-iedg 26891  df-edg 26940  df-uhgr 26950  df-upgr 26974  df-umgr 26975  df-usgr 27043 This theorem is referenced by:  fusgrfis  27219  cusgrres  27337
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