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Theorem frgrncvvdeqlem2 27482
Description: Lemma 2 for frgrncvvdeq 27491. 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,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   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 466 . . 3 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2842 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrncvvdeq.v1 . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65nbgrisvtx 26458 . . . . . . 7 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
76a1i 11 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
84, 7syl5bi 232 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
98imp 393 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
10 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1110adantr 466 . . . 4 ((𝜑𝑥𝐷) → 𝑌𝑉)
12 frgrncvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
13 elnelne2 3057 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1413expcom 398 . . . . . 6 (𝑌𝐷 → (𝑥𝐷𝑥𝑌))
1512, 14syl 17 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑌))
1615imp 393 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
179, 11, 163jca 1122 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
18 frgrncvvdeq.e . . . 4 𝐸 = (Edg‘𝐺)
195, 18frcond1 27448 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
202, 17, 19sylc 65 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21 frgrusgr 27442 . . . 4 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22 usgrumgr 26296 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
235, 18umgrpredgv 26257 . . . . . . . . . . . . . 14 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑥𝑉𝑦𝑉))
2423simprd 483 . . . . . . . . . . . . 13 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉)
2524ex 397 . . . . . . . . . . . 12 (𝐺 ∈ UMGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
2622, 25syl 17 . . . . . . . . . . 11 (𝐺 ∈ USGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
2726adantld 478 . . . . . . . . . 10 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉))
2827pm4.71rd 552 . . . . . . . . 9 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))))
29 prex 5037 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
30 prex 5037 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
3129, 30prss 4486 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
32 ancom 452 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
3331, 32bitr3i 266 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
3433anbi2i 609 . . . . . . . . 9 ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
3528, 34syl6rbbr 279 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
36 frgrncvvdeq.ny . . . . . . . . . . 11 𝑁 = (𝐺 NeighbVtx 𝑌)
3736eleq2i 2842 . . . . . . . . . 10 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
3818nbusgreledg 26472 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
3937, 38syl5rbb 273 . . . . . . . . 9 (𝐺 ∈ USGraph → ({𝑦, 𝑌} ∈ 𝐸𝑦𝑁))
4039anbi1d 615 . . . . . . . 8 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4135, 40bitrd 268 . . . . . . 7 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4241eubidv 2638 . . . . . 6 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4342biimpd 219 . . . . 5 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44 df-reu 3068 . . . . 5 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45 df-reu 3068 . . . . 5 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
4643, 44, 453imtr4g 285 . . . 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 466 . 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 382  w3a 1071   = wceq 1631  wcel 2145  ∃!weu 2618  wne 2943  wnel 3046  ∃!wreu 3063  wss 3723  {cpr 4318  cmpt 4863  cfv 6031  crio 6753  (class class class)co 6793  Vtxcvtx 26095  Edgcedg 26160  UMGraphcumgr 26197  USGraphcusgr 26266   NeighbVtx cnbgr 26447   FriendGraph cfrgr 27438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-hash 13322  df-edg 26161  df-upgr 26198  df-umgr 26199  df-usgr 26268  df-nbgr 26448  df-frgr 27439
This theorem is referenced by:  frgrncvvdeqlem3  27483  frgrncvvdeqlem4  27484
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