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Theorem upgrres1 28570
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 28525 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 6872 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of 6834 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹𝐹)
43ffdmd 6749 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
5 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
6 simpr 486 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒𝐸)
76adantr 482 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝐸)
8 upgrres1.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
98eleq2i 2826 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
10 edgupgr 28394 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
11 elpwi 4610 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
12 upgrres1.v . . . . . . . . . . . . . . 15 𝑉 = (Vtx‘𝐺)
1311, 12sseqtrrdi 4034 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒𝑉)
14133ad2ant1 1134 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2) → 𝑒𝑉)
1510, 14syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒𝑉)
169, 15sylan2b 595 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ 𝑒𝐸) → 𝑒𝑉)
1716ad4ant13 750 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝑉)
18 simpr 486 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑁𝑒)
19 elpwdifsn 4793 . . . . . . . . . 10 ((𝑒𝐸𝑒𝑉𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
207, 17, 18, 19syl3anc 1372 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
21 simpl 484 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐺 ∈ UPGraph)
229biimpi 215 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
2310simp2d 1144 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
2421, 22, 23syl2an 597 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒 ≠ ∅)
2524adantr 482 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ≠ ∅)
26 nelsn 4669 . . . . . . . . . 10 (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
2725, 26syl 17 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → ¬ 𝑒 ∈ {∅})
2820, 27eldifd 3960 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
2928ex 414 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3029ralrimiva 3147 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
31 rabss 4070 . . . . . 6 ({𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3230, 31sylibr 233 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
335, 32eqsstrid 4031 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
34 elrabi 3678 . . . . . . 7 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → 𝑝𝐸)
35 edgval 28309 . . . . . . . . . . . 12 (Edg‘𝐺) = ran (iEdg‘𝐺)
368, 35eqtri 2761 . . . . . . . . . . 11 𝐸 = ran (iEdg‘𝐺)
3736eleq2i 2826 . . . . . . . . . 10 (𝑝𝐸𝑝 ∈ ran (iEdg‘𝐺))
38 eqid 2733 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
3912, 38upgrf 28346 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4039frnd 6726 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4140sseld 3982 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4237, 41biimtrid 241 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43 fveq2 6892 . . . . . . . . . . . 12 (𝑥 = 𝑝 → (♯‘𝑥) = (♯‘𝑝))
4443breq1d 5159 . . . . . . . . . . 11 (𝑥 = 𝑝 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑝) ≤ 2))
4544elrab 3684 . . . . . . . . . 10 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝑝) ≤ 2))
4645simprbi 498 . . . . . . . . 9 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝑝) ≤ 2)
4742, 46syl6 35 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝑝𝐸 → (♯‘𝑝) ≤ 2))
4847adantr 482 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑝𝐸 → (♯‘𝑝) ≤ 2))
4934, 48syl5com 31 . . . . . 6 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (♯‘𝑝) ≤ 2))
5049, 5eleq2s 2852 . . . . 5 (𝑝𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (♯‘𝑝) ≤ 2))
5150impcom 409 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑝𝐹) → (♯‘𝑝) ≤ 2)
5233, 51ssrabdv 4072 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
534, 52fssd 6736 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
54 upgrres1.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
55 opex 5465 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
5654, 55eqeltri 2830 . . 3 𝑆 ∈ V
5712, 8, 5, 54upgrres1lem2 28568 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
5857eqcomi 2742 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
5912, 8, 5, 54upgrres1lem3 28569 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
6059eqcomi 2742 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
6158, 60isupgr 28344 . . 3 (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
6256, 61mp1i 13 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑆 ∈ UPGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶{𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
6353, 62mpbird 257 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wnel 3047  wral 3062  {crab 3433  Vcvv 3475  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629  cop 4635   class class class wbr 5149   I cid 5574  dom cdm 5677  ran crn 5678  cres 5679  wf 6540  1-1-ontowf1o 6543  cfv 6544  cle 11249  2c2 12267  chash 14290  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307  UPGraphcupgr 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976  df-vtx 28258  df-iedg 28259  df-edg 28308  df-upgr 28342
This theorem is referenced by:  nbupgrres  28621
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