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Theorem frgrncvvdeqlem2 30319
Description: Lemma 2 for frgrncvvdeq 30328. In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-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
frgrncvvdeqlem2 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑁(𝑥,𝑦)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem2
StepHypRef Expression
1 frgrncvvdeq.f . . . 4 (𝜑𝐺 ∈ FriendGraph )
21adantr 480 . . 3 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2833 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrncvvdeq.v1 . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65nbgrisvtx 29358 . . . . . . 7 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
76a1i 11 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
84, 7biimtrid 242 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
98imp 406 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
10 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1110adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑌𝑉)
12 frgrncvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
13 elnelne2 3058 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1413expcom 413 . . . . . 6 (𝑌𝐷 → (𝑥𝐷𝑥𝑌))
1512, 14syl 17 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑌))
1615imp 406 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
179, 11, 163jca 1129 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
18 frgrncvvdeq.e . . . 4 𝐸 = (Edg‘𝐺)
195, 18frcond1 30285 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
202, 17, 19sylc 65 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21 frgrusgr 30280 . . . 4 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22 prex 5437 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
23 prex 5437 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
2422, 23prss 4820 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
25 ancom 460 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
2624, 25bitr3i 277 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
2726anbi2i 623 . . . . . . . . 9 ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
28 usgrumgr 29198 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
295, 18umgrpredgv 29157 . . . . . . . . . . . . . 14 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑥𝑉𝑦𝑉))
3029simprd 495 . . . . . . . . . . . . 13 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉)
3130ex 412 . . . . . . . . . . . 12 (𝐺 ∈ UMGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
3228, 31syl 17 . . . . . . . . . . 11 (𝐺 ∈ USGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
3332adantld 490 . . . . . . . . . 10 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉))
3433pm4.71rd 562 . . . . . . . . 9 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))))
3527, 34bitr4id 290 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
36 frgrncvvdeq.ny . . . . . . . . . . 11 𝑁 = (𝐺 NeighbVtx 𝑌)
3736eleq2i 2833 . . . . . . . . . 10 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
3818nbusgreledg 29370 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
3937, 38bitr2id 284 . . . . . . . . 9 (𝐺 ∈ USGraph → ({𝑦, 𝑌} ∈ 𝐸𝑦𝑁))
4039anbi1d 631 . . . . . . . 8 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4135, 40bitrd 279 . . . . . . 7 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4241eubidv 2586 . . . . . 6 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4342biimpd 229 . . . . 5 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44 df-reu 3381 . . . . 5 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45 df-reu 3381 . . . . 5 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
4643, 44, 453imtr4g 296 . . . 4 (𝐺 ∈ USGraph → (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
471, 21, 463syl 18 . . 3 (𝜑 → (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
4847adantr 480 . 2 ((𝜑𝑥𝐷) → (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
4920, 48mpd 15 1 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ∃!weu 2568  wne 2940  wnel 3046  ∃!wreu 3378  wss 3951  {cpr 4628  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064  UMGraphcumgr 29098  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:  frgrncvvdeqlem3  30320  frgrncvvdeqlem4  30321
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