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Theorem frgrncvvdeqlem3 30320
Description: Lemma 3 for frgrncvvdeq 30328. 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,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   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 4217 . 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 2833 . . . . . . 7 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
87nbgrisvtx 29358 . . . . . . . 8 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
98a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
106, 9biimtrid 242 . . . . . 6 (𝜑 → (𝑥𝐷𝑥𝑉))
1110imp 406 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑉)
12 frgrncvvdeq.y . . . . . 6 (𝜑𝑌𝑉)
1312adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝑌𝑉)
14 frgrncvvdeq.xy . . . . . . 7 (𝜑𝑌𝐷)
15 elnelne2 3058 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1614, 15sylan2 593 . . . . . 6 ((𝑥𝐷𝜑) → 𝑥𝑌)
1716ancoms 458 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑌)
1811, 13, 173jca 1129 . . . 4 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
19 frgrncvvdeq.e . . . . 5 𝐸 = (Edg‘𝐺)
207, 19frcond3 30288 . . . 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 3484 . . . . . . . . . 10 𝑛 ∈ V
23 elinsn 4710 . . . . . . . . . 10 ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
2422, 23mpan 690 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25 frgrusgr 30280 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2619nbusgreledg 29370 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27 prcom 4732 . . . . . . . . . . . . . . . . . 18 {𝑛, 𝑥} = {𝑥, 𝑛}
2827eleq1i 2832 . . . . . . . . . . . . . . . . 17 ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
2926, 28bitrdi 287 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
3029biimpd 229 . . . . . . . . . . . . . . 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 406 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
361eqcomi 2746 . . . . . . . . . . . . . 14 (𝐺 NeighbVtx 𝑌) = 𝑁
3736eleq2i 2833 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛𝑁)
3837biimpi 216 . . . . . . . . . . . 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 30319 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
44 preq2 4734 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
4544eleq1d 2826 . . . . . . . . . . . 12 (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
4645riota2 7413 . . . . . . . . . . 11 ((𝑛𝑁 ∧ ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4739, 43, 46syl2an 596 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4835, 47mpbid 232 . . . . . . . . 9 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
4924, 48sylan 580 . . . . . . . 8 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
5049eqcomd 2743 . . . . . . 7 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → 𝑛 = (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
5150sneqd 4638 . . . . . 6 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52 eqeq1 2741 . . . . . . 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 257 . . . . 5 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
5554ex 412 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5655rexlimivw 3151 . . 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, 57eqtr2id 2790 1 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wrex 3070  ∃!wreu 3378  Vcvv 3480  cin 3950  {csn 4626  {cpr 4628  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166   NeighbVtx cnbgr 29349   FriendGraph cfrgr 30277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-edg 29065  df-upgr 29099  df-umgr 29100  df-usgr 29168  df-nbgr 29350  df-frgr 30278
This theorem is referenced by:  frgrncvvdeqlem5  30322
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