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Theorem frgrncvvdeqlem2 27580
Description: Lemma 2 for frgrncvvdeq 27589. 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 472 . . 3 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2836 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrncvvdeq.v1 . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65nbgrisvtx 26514 . . . . . . 7 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
76a1i 11 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
84, 7syl5bi 233 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
98imp 395 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
10 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1110adantr 472 . . . 4 ((𝜑𝑥𝐷) → 𝑌𝑉)
12 frgrncvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
13 elnelne2 3051 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1413expcom 402 . . . . . 6 (𝑌𝐷 → (𝑥𝐷𝑥𝑌))
1512, 14syl 17 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑌))
1615imp 395 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
179, 11, 163jca 1158 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
18 frgrncvvdeq.e . . . 4 𝐸 = (Edg‘𝐺)
195, 18frcond1 27546 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
202, 17, 19sylc 65 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21 frgrusgr 27540 . . . 4 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22 usgrumgr 26352 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
235, 18umgrpredgv 26313 . . . . . . . . . . . . . 14 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑥𝑉𝑦𝑉))
2423simprd 489 . . . . . . . . . . . . 13 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉)
2524ex 401 . . . . . . . . . . . 12 (𝐺 ∈ UMGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
2622, 25syl 17 . . . . . . . . . . 11 (𝐺 ∈ USGraph → ({𝑥, 𝑦} ∈ 𝐸𝑦𝑉))
2726adantld 484 . . . . . . . . . 10 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉))
2827pm4.71rd 558 . . . . . . . . 9 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))))
29 prex 5065 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
30 prex 5065 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
3129, 30prss 4505 . . . . . . . . . . 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 268 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
3433anbi2i 616 . . . . . . . . 9 ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
3528, 34syl6rbbr 281 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
36 frgrncvvdeq.ny . . . . . . . . . . 11 𝑁 = (𝐺 NeighbVtx 𝑌)
3736eleq2i 2836 . . . . . . . . . 10 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
3818nbusgreledg 26528 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
3937, 38syl5rbb 275 . . . . . . . . 9 (𝐺 ∈ USGraph → ({𝑦, 𝑌} ∈ 𝐸𝑦𝑁))
4039anbi1d 623 . . . . . . . 8 (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4135, 40bitrd 270 . . . . . . 7 (𝐺 ∈ USGraph → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4241eubidv 2585 . . . . . 6 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
4342biimpd 220 . . . . 5 (𝐺 ∈ USGraph → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44 df-reu 3062 . . . . 5 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45 df-reu 3062 . . . . 5 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
4643, 44, 453imtr4g 287 . . . 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 472 . 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 384  w3a 1107   = wceq 1652  wcel 2155  ∃!weu 2581  wne 2937  wnel 3040  ∃!wreu 3057  wss 3732  {cpr 4336  cmpt 4888  cfv 6068  crio 6802  (class class class)co 6842  Vtxcvtx 26165  Edgcedg 26216  UMGraphcumgr 26253  USGraphcusgr 26322   NeighbVtx cnbgr 26503   FriendGraph cfrgr 27536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-edg 26217  df-upgr 26254  df-umgr 26255  df-usgr 26324  df-nbgr 26504  df-frgr 27537
This theorem is referenced by:  frgrncvvdeqlem3  27581  frgrncvvdeqlem4  27582
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