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Theorem frgrncvvdeqlem2 30329
Description: Lemma 2 for frgrncvvdeq 30338. 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 2831 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrncvvdeq.v1 . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65nbgrisvtx 29373 . . . . . . 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 3056 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1413expcom 413 . . . . . 6 (𝑌𝐷 → (𝑥𝐷𝑥𝑌))
1512, 14syl 17 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑌))
1615imp 406 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
179, 11, 163jca 1127 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
18 frgrncvvdeq.e . . . 4 𝐸 = (Edg‘𝐺)
195, 18frcond1 30295 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
202, 17, 19sylc 65 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21 frgrusgr 30290 . . . 4 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22 prex 5443 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
23 prex 5443 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
2422, 23prss 4825 . . . . . . . . . . 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 29213 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
295, 18umgrpredgv 29172 . . . . . . . . . . . . . 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 2831 . . . . . . . . . 10 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
3818nbusgreledg 29385 . . . . . . . . . 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 2584 . . . . . 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 3379 . . . . 5 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45 df-reu 3379 . . . . 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 1086   = wceq 1537  wcel 2106  ∃!weu 2566  wne 2938  wnel 3044  ∃!wreu 3376  wss 3963  {cpr 4633  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079  UMGraphcumgr 29113  USGraphcusgr 29181   NeighbVtx cnbgr 29364   FriendGraph cfrgr 30287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-upgr 29114  df-umgr 29115  df-usgr 29183  df-nbgr 29365  df-frgr 30288
This theorem is referenced by:  frgrncvvdeqlem3  30330  frgrncvvdeqlem4  30331
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