Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  upgrres1 Structured version   Visualization version   GIF version

Theorem upgrres1 40532
 Description: A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This 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 AV, 8-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
upgrres1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph )
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem upgrres1
Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 6086 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of 6050 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹𝐹)
4 dmresi 5376 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
54a1i 11 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
65feq2d 5944 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹𝐹))
73, 6mpbird 246 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
8 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
9 simpr 476 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒𝐸)
109adantr 480 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝐸)
11 upgrres1.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
1211eleq2i 2680 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edgupgr 25808 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (#‘𝑒) ≤ 2))
14 elpwi 4117 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
15 upgrres1.v . . . . . . . . . . . . . . 15 𝑉 = (Vtx‘𝐺)
1614, 15syl6sseqr 3615 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒𝑉)
17163ad2ant1 1075 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (#‘𝑒) ≤ 2) → 𝑒𝑉)
1813, 17syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒𝑉)
1912, 18sylan2b 491 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ 𝑒𝐸) → 𝑒𝑉)
2019ad4ant13 1284 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝑉)
21 simpr 476 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑁𝑒)
22 elpwdifsn 40312 . . . . . . . . . 10 ((𝑒𝐸𝑒𝑉𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
2310, 20, 21, 22syl3anc 1318 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
24 simpl 472 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐺 ∈ UPGraph )
2512biimpi 205 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
2613simp2d 1067 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
2724, 25, 26syl2an 493 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒 ≠ ∅)
2827adantr 480 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ≠ ∅)
29 nelsn 4159 . . . . . . . . . 10 (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
3028, 29syl 17 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → ¬ 𝑒 ∈ {∅})
3123, 30eldifd 3551 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
3231ex 449 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3332ralrimiva 2949 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
34 rabss 3642 . . . . . 6 ({𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3533, 34sylibr 223 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
368, 35syl5eqss 3612 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
37 elrabi 3328 . . . . . . 7 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → 𝑝𝐸)
38 edgaval 25794 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3911, 38syl5eq 2656 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4039eleq2d 2673 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ ran (iEdg‘𝐺)))
41 eqid 2610 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
4215, 41upgrf 25753 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
43 frn 5966 . . . . . . . . . . . 12 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4442, 43syl 17 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4544sseld 3567 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
4640, 45sylbid 229 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
47 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝑝 → (#‘𝑥) = (#‘𝑝))
4847breq1d 4593 . . . . . . . . . . 11 (𝑥 = 𝑝 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑝) ≤ 2))
4948elrab 3331 . . . . . . . . . 10 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝑝) ≤ 2))
5049simprbi 479 . . . . . . . . 9 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (#‘𝑝) ≤ 2)
5146, 50syl6 34 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝑝𝐸 → (#‘𝑝) ≤ 2))
5251adantr 480 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑝𝐸 → (#‘𝑝) ≤ 2))
5337, 52syl5com 31 . . . . . 6 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (#‘𝑝) ≤ 2))
5453, 8eleq2s 2706 . . . . 5 (𝑝𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (#‘𝑝) ≤ 2))
5554impcom 445 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑝𝐹) → (#‘𝑝) ≤ 2)
5636, 55ssrabdv 3644 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
577, 56fssd 5970 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
58 upgrres1.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
59 opex 4859 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
6058, 59eqeltri 2684 . . 3 𝑆 ∈ V
6115, 11, 8, 58upgrres1lem2 40530 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
6261eqcomi 2619 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
6315, 11, 8, 58upgrres1lem3 40531 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
6463eqcomi 2619 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
6562, 64isupgr 25751 . . 3 (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
6660, 65mp1i 13 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
6757, 66mpbird 246 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131   class class class wbr 4583   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792 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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-vtx 25675  df-iedg 25676  df-upgr 25749  df-edga 25793 This theorem is referenced by:  nbupgrres  40592
 Copyright terms: Public domain W3C validator