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Theorem upgrres1 40527
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 40487 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 6071 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of 6035 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹𝐹)
4 dmresi 5363 . . . . . 6 dom ( I ↾ 𝐹) = 𝐹
54a1i 11 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → dom ( I ↾ 𝐹) = 𝐹)
65feq2d 5930 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹 ↔ ( I ↾ 𝐹):𝐹𝐹))
73, 6mpbird 245 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
8 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
9 simpr 475 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒𝐸)
109adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝐸)
11 upgrres1.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
1211eleq2i 2679 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edgupgr 40362 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (#‘𝑒) ≤ 2))
14 elpwi 4116 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
15 upgrres1.v . . . . . . . . . . . . . . 15 𝑉 = (Vtx‘𝐺)
1614, 15syl6sseqr 3614 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒𝑉)
17163ad2ant1 1074 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (#‘𝑒) ≤ 2) → 𝑒𝑉)
1813, 17syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒𝑉)
1912, 18sylan2b 490 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ 𝑒𝐸) → 𝑒𝑉)
2019ad4ant13 1283 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝑉)
21 simpr 475 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑁𝑒)
22 elpwdifsn 40110 . . . . . . . . . 10 ((𝑒𝐸𝑒𝑉𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
2310, 20, 21, 22syl3anc 1317 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
24 simpl 471 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐺 ∈ UPGraph )
2512biimpi 204 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
2613simp2d 1066 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
2724, 25, 26syl2an 492 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒 ≠ ∅)
2827adantr 479 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ≠ ∅)
29 nelsn 4158 . . . . . . . . . 10 (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
3028, 29syl 17 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → ¬ 𝑒 ∈ {∅})
3123, 30eldifd 3550 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
3231ex 448 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3332ralrimiva 2948 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
34 rabss 3641 . . . . . 6 ({𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3533, 34sylibr 222 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
368, 35syl5eqss 3611 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
37 elrabi 3327 . . . . . . 7 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → 𝑝𝐸)
38 edgaval 40348 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3911, 38syl5eq 2655 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4039eleq2d 2672 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ ran (iEdg‘𝐺)))
41 eqid 2609 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
4215, 41upgrf 40307 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
43 frn 5952 . . . . . . . . . . . 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 3566 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
4640, 45sylbid 228 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
47 fveq2 6088 . . . . . . . . . . . 12 (𝑥 = 𝑝 → (#‘𝑥) = (#‘𝑝))
4847breq1d 4587 . . . . . . . . . . 11 (𝑥 = 𝑝 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑝) ≤ 2))
4948elrab 3330 . . . . . . . . . 10 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝑝) ≤ 2))
5049simprbi 478 . . . . . . . . 9 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (#‘𝑝) ≤ 2)
5146, 50syl6 34 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝑝𝐸 → (#‘𝑝) ≤ 2))
5251adantr 479 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑝𝐸 → (#‘𝑝) ≤ 2))
5337, 52syl5com 31 . . . . . 6 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (#‘𝑝) ≤ 2))
5453, 8eleq2s 2705 . . . . 5 (𝑝𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (#‘𝑝) ≤ 2))
5554impcom 444 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑝𝐹) → (#‘𝑝) ≤ 2)
5636, 55ssrabdv 3643 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
577, 56fssd 5956 . 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 4853 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
6058, 59eqeltri 2683 . . 3 𝑆 ∈ V
6115, 11, 8, 58upgrres1lem2 40525 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
6261eqcomi 2618 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
6315, 11, 8, 58upgrres1lem3 40526 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
6463eqcomi 2618 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
6562, 64isupgr 40305 . . 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 245 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  {crab 2899  Vcvv 3172  cdif 3536  wss 3539  c0 3873  𝒫 cpw 4107  {csn 4124  cop 4130   class class class wbr 4577   I cid 4938  dom cdm 5028  ran crn 5029  cres 5030  wf 5786  1-1-ontowf1o 5789  cfv 5790  cle 9931  2c2 10917  #chash 12934  Vtxcvtx 40224  iEdgciedg 40225   UPGraph cupgr 40301  Edgcedga 40346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-1st 7036  df-2nd 7037  df-vtx 40226  df-iedg 40227  df-upgr 40303  df-edga 40347
This theorem is referenced by:  nbupgrres  40587
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