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Theorem frgrncvvdeqlem3 27281
Description: Lemma 3 for frgrncvvdeq 27289. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem3 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Distinct variable groups:   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   𝑦,𝐸   𝑦,𝑁
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
21ineq2i 3844 . 2 ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))
3 frgrncvvdeq.f . . . . 5 (𝜑𝐺 ∈ FriendGraph )
43adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
5 frgrncvvdeq.nx . . . . . . . 8 𝐷 = (𝐺 NeighbVtx 𝑋)
65eleq2i 2722 . . . . . . 7 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
87nbgrisvtx 26280 . . . . . . . 8 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
98a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
106, 9syl5bi 232 . . . . . 6 (𝜑 → (𝑥𝐷𝑥𝑉))
1110imp 444 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑉)
12 frgrncvvdeq.y . . . . . 6 (𝜑𝑌𝑉)
1312adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝑌𝑉)
14 frgrncvvdeq.xy . . . . . . 7 (𝜑𝑌𝐷)
15 elnelne2 2937 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1614, 15sylan2 490 . . . . . 6 ((𝑥𝐷𝜑) → 𝑥𝑌)
1716ancoms 468 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑌)
1811, 13, 173jca 1261 . . . 4 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
19 frgrncvvdeq.e . . . . 5 𝐸 = (Edg‘𝐺)
207, 19frcond3 27249 . . . 4 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
214, 18, 20sylc 65 . . 3 ((𝜑𝑥𝐷) → ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛})
22 vex 3234 . . . . . . . . . 10 𝑛 ∈ V
23 elinsn 4277 . . . . . . . . . 10 ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
2422, 23mpan 706 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25 frgrusgr 27240 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2619nbusgreledg 26294 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27 prcom 4299 . . . . . . . . . . . . . . . . . 18 {𝑛, 𝑥} = {𝑥, 𝑛}
2827eleq1i 2721 . . . . . . . . . . . . . . . . 17 ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
2926, 28syl6bb 276 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
3029biimpd 219 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
313, 25, 303syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
3231adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
3332com12 32 . . . . . . . . . . . 12 (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
3433adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
3534imp 444 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
361eqcomi 2660 . . . . . . . . . . . . . 14 (𝐺 NeighbVtx 𝑌) = 𝑁
3736eleq2i 2722 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛𝑁)
3837biimpi 206 . . . . . . . . . . . 12 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑁)
3938adantl 481 . . . . . . . . . . 11 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛𝑁)
40 frgrncvvdeq.x . . . . . . . . . . . 12 (𝜑𝑋𝑉)
41 frgrncvvdeq.ne . . . . . . . . . . . 12 (𝜑𝑋𝑌)
42 frgrncvvdeq.a . . . . . . . . . . . 12 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
437, 19, 5, 1, 40, 12, 41, 14, 3, 42frgrncvvdeqlem2 27280 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
44 preq2 4301 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
4544eleq1d 2715 . . . . . . . . . . . 12 (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
4645riota2 6673 . . . . . . . . . . 11 ((𝑛𝑁 ∧ ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4739, 43, 46syl2an 493 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4835, 47mpbid 222 . . . . . . . . 9 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
4924, 48sylan 487 . . . . . . . 8 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
5049eqcomd 2657 . . . . . . 7 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → 𝑛 = (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
5150sneqd 4222 . . . . . 6 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52 eqeq1 2655 . . . . . . 7 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5352adantr 480 . . . . . 6 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5451, 53mpbird 247 . . . . 5 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
5554ex 449 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5655rexlimivw 3058 . . 3 (∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5721, 56mpcom 38 . 2 ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
582, 57syl5req 2698 1 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wrex 2942  ∃!wreu 2943  Vcvv 3231  cin 3606  {csn 4210  {cpr 4212  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269   FriendGraph cfrgr 27236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-nbgr 26270  df-frgr 27237
This theorem is referenced by:  frgrncvvdeqlem5  27283
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