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Theorem usgrres1 40534
 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 40492 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 6086 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of1 6049 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹1-1𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹1-1𝐹)
4 eqidd 2611 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5 dmresi 5376 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
65a1i 11 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7 eqidd 2611 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹 = 𝐹)
84, 6, 7f1eq123d 6044 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹 ↔ ( I ↾ 𝐹):𝐹1-1𝐹))
93, 8mpbird 246 . . 3 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1𝐹)
10 usgrumgr 40409 . . . 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 40529 . . . 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 6020 . . 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 691 . 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 4859 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
2018, 19eqeltri 2684 . . 3 𝑆 ∈ V
2111, 12, 13, 18upgrres1lem2 40530 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
2221eqcomi 2619 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
2311, 12, 13, 18upgrres1lem3 40531 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
2423eqcomi 2619 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
2522, 24isusgrs 40386 . . 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 246 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝑆 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟨cop 4131   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UMGraph cumgr 25748  Edgcedga 25792   USGraph cusgr 40379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381 This theorem is referenced by:  fusgrfis  40549  cusgrres  40664
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