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Theorem upgrres1 27103
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 27058 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 6627 . . . . 5 ( I ↾ 𝐹):𝐹1-1-onto𝐹
2 f1of 6590 . . . . 5 (( I ↾ 𝐹):𝐹1-1-onto𝐹 → ( I ↾ 𝐹):𝐹𝐹)
31, 2mp1i 13 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):𝐹𝐹)
43ffdmd 6511 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)⟶𝐹)
5 upgrres1.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
6 simpr 488 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒𝐸)
76adantr 484 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝐸)
8 upgrres1.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
98eleq2i 2881 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
10 edgupgr 26927 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
11 elpwi 4506 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒 ⊆ (Vtx‘𝐺))
12 upgrres1.v . . . . . . . . . . . . . . 15 𝑉 = (Vtx‘𝐺)
1311, 12sseqtrrdi 3966 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 (Vtx‘𝐺) → 𝑒𝑉)
14133ad2ant1 1130 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2) → 𝑒𝑉)
1510, 14syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒𝑉)
169, 15sylan2b 596 . . . . . . . . . . 11 ((𝐺 ∈ UPGraph ∧ 𝑒𝐸) → 𝑒𝑉)
1716ad4ant13 750 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒𝑉)
18 simpr 488 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑁𝑒)
19 elpwdifsn 4682 . . . . . . . . . 10 ((𝑒𝐸𝑒𝑉𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
207, 17, 18, 19syl3anc 1368 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}))
21 simpl 486 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐺 ∈ UPGraph)
229biimpi 219 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
2310simp2d 1140 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≠ ∅)
2421, 22, 23syl2an 598 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → 𝑒 ≠ ∅)
2524adantr 484 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ≠ ∅)
26 nelsn 4565 . . . . . . . . . 10 (𝑒 ≠ ∅ → ¬ 𝑒 ∈ {∅})
2725, 26syl 17 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → ¬ 𝑒 ∈ {∅})
2820, 27eldifd 3892 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) ∧ 𝑁𝑒) → 𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
2928ex 416 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑒𝐸) → (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3029ralrimiva 3149 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
31 rabss 3999 . . . . . 6 ({𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ↔ ∀𝑒𝐸 (𝑁𝑒𝑒 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅})))
3230, 31sylibr 237 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
335, 32eqsstrid 3963 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}))
34 elrabi 3623 . . . . . . 7 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → 𝑝𝐸)
35 edgval 26842 . . . . . . . . . . . 12 (Edg‘𝐺) = ran (iEdg‘𝐺)
368, 35eqtri 2821 . . . . . . . . . . 11 𝐸 = ran (iEdg‘𝐺)
3736eleq2i 2881 . . . . . . . . . 10 (𝑝𝐸𝑝 ∈ ran (iEdg‘𝐺))
38 eqid 2798 . . . . . . . . . . . . 13 (iEdg‘𝐺) = (iEdg‘𝐺)
3912, 38upgrf 26879 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4039frnd 6494 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4140sseld 3914 . . . . . . . . . 10 (𝐺 ∈ UPGraph → (𝑝 ∈ ran (iEdg‘𝐺) → 𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4237, 41syl5bi 245 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑝𝐸𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43 fveq2 6645 . . . . . . . . . . . 12 (𝑥 = 𝑝 → (♯‘𝑥) = (♯‘𝑝))
4443breq1d 5040 . . . . . . . . . . 11 (𝑥 = 𝑝 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑝) ≤ 2))
4544elrab 3628 . . . . . . . . . 10 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝑝) ≤ 2))
4645simprbi 500 . . . . . . . . 9 (𝑝 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘𝑝) ≤ 2)
4742, 46syl6 35 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝑝𝐸 → (♯‘𝑝) ≤ 2))
4847adantr 484 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝑝𝐸 → (♯‘𝑝) ≤ 2))
4934, 48syl5com 31 . . . . . 6 (𝑝 ∈ {𝑒𝐸𝑁𝑒} → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (♯‘𝑝) ≤ 2))
5049, 5eleq2s 2908 . . . . 5 (𝑝𝐹 → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (♯‘𝑝) ≤ 2))
5150impcom 411 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑝𝐹) → (♯‘𝑝) ≤ 2)
5233, 51ssrabdv 4001 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝐹 ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
534, 52fssd 6502 . 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 5321 . . . 4 ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
5654, 55eqeltri 2886 . . 3 𝑆 ∈ V
5712, 8, 5, 54upgrres1lem2 27101 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
5857eqcomi 2807 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
5912, 8, 5, 54upgrres1lem3 27102 . . . . 5 (iEdg‘𝑆) = ( I ↾ 𝐹)
6059eqcomi 2807 . . . 4 ( I ↾ 𝐹) = (iEdg‘𝑆)
6158, 60isupgr 26877 . . 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 260 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  wnel 3091  wral 3106  {crab 3110  Vcvv 3441  cdif 3878  wss 3881  c0 4243  𝒫 cpw 4497  {csn 4525  cop 4531   class class class wbr 5030   I cid 5424  dom cdm 5519  ran crn 5520  cres 5521  wf 6320  1-1-ontowf1o 6323  cfv 6324  cle 10665  2c2 11680  chash 13686  Vtxcvtx 26789  iEdgciedg 26790  Edgcedg 26840  UPGraphcupgr 26873
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 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1st 7671  df-2nd 7672  df-vtx 26791  df-iedg 26792  df-edg 26841  df-upgr 26875
This theorem is referenced by:  nbupgrres  27154
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