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Theorem frgrncvvdeqlem3 27063
Description: Lemma 3 for frgrncvvdeq 27072. In a friendship graph, for each neighbor of a vertex there is exacly 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.)
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)
Distinct variable groups:   𝑦,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐷(𝑥)   𝐸(𝑥,𝑦)   𝐺(𝑥)   𝑁(𝑥,𝑦)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem3
StepHypRef Expression
1 frgrncvvdeq.f . . . 4 (𝜑𝐺 ∈ FriendGraph )
21adantr 481 . . 3 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2690 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrusgr 27024 . . . . . . 7 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
6 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
76nbgrisvtx 26176 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑥 ∈ (𝐺 NeighbVtx 𝑋)) → 𝑥𝑉)
87ex 450 . . . . . . 7 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
91, 5, 83syl 18 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
104, 9syl5bi 232 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
1110imp 445 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
12 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1312adantr 481 . . . 4 ((𝜑𝑥𝐷) → 𝑌𝑉)
14 frgrncvvdeq.e . . . . . 6 𝐸 = (Edg‘𝐺)
15 frgrncvvdeq.ny . . . . . 6 𝑁 = (𝐺 NeighbVtx 𝑌)
16 frgrncvvdeq.x . . . . . 6 (𝜑𝑋𝑉)
17 frgrncvvdeq.ne . . . . . 6 (𝜑𝑋𝑌)
18 frgrncvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
19 frgrncvvdeq.a . . . . . 6 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
206, 14, 3, 15, 16, 12, 17, 18, 1, 19frgrncvvdeqlem1 27061 . . . . 5 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
21 eldif 3570 . . . . . 6 (𝑌 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑌𝑉 ∧ ¬ 𝑌 ∈ {𝑥}))
22 vsnid 4187 . . . . . . . 8 𝑥 ∈ {𝑥}
23 eleq1 2686 . . . . . . . . 9 (𝑌 = 𝑥 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2423eqcoms 2629 . . . . . . . 8 (𝑥 = 𝑌 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2522, 24mpbiri 248 . . . . . . 7 (𝑥 = 𝑌𝑌 ∈ {𝑥})
2625necon3bi 2816 . . . . . 6 𝑌 ∈ {𝑥} → 𝑥𝑌)
2721, 26simplbiim 658 . . . . 5 (𝑌 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑌)
2820, 27syl 17 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
2911, 13, 283jca 1240 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
306, 14frcond1 27030 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
312, 29, 30sylc 65 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
32 prex 4880 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
33 prex 4880 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
3432, 33prss 4326 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
35 simpr 477 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) → {𝑦, 𝑌} ∈ 𝐸)
3634, 35sylbir 225 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → {𝑦, 𝑌} ∈ 𝐸)
3736ad2antll 764 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → {𝑦, 𝑌} ∈ 𝐸)
3815a1i 11 . . . . . . . . . . . . 13 (𝜑𝑁 = (𝐺 NeighbVtx 𝑌))
3938eleq2d 2684 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌)))
4014nbusgreledg 26170 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
411, 5, 403syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
4239, 41bitrd 268 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4342adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4443adantr 481 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4537, 44mpbird 247 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → 𝑦𝑁)
46 simpl 473 . . . . . . . . . 10 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
4734, 46sylbir 225 . . . . . . . . 9 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → {𝑥, 𝑦} ∈ 𝐸)
4847ad2antll 764 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → {𝑥, 𝑦} ∈ 𝐸)
4945, 48jca 554 . . . . . . 7 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
5049ex 450 . . . . . 6 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
5115eleq2i 2690 . . . . . . . . . . . . 13 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
5251, 41syl5bb 272 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
5352biimpd 219 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 → {𝑦, 𝑌} ∈ 𝐸))
5453adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 → {𝑦, 𝑌} ∈ 𝐸))
5554impcom 446 . . . . . . . . 9 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → {𝑦, 𝑌} ∈ 𝐸)
566nbgrisvtx 26176 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑦𝑉)
5756ex 450 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) → 𝑦𝑉))
581, 5, 573syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) → 𝑦𝑉))
5951, 58syl5bi 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑁𝑦𝑉))
6059adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → (𝑦𝑁𝑦𝑉))
6160impcom 446 . . . . . . . . . . . 12 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → 𝑦𝑉)
6261ad2antlr 762 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉)
63 simpl 473 . . . . . . . . . . . . 13 (({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → {𝑦, 𝑌} ∈ 𝐸)
64 id 22 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ∈ 𝐸 → {𝑥, 𝑦} ∈ 𝐸)
6563, 64anim12ci 590 . . . . . . . . . . . 12 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸))
6665, 34sylib 208 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
6762, 66jca 554 . . . . . . . . . 10 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
6867ex 450 . . . . . . . . 9 (({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
6955, 68mpancom 702 . . . . . . . 8 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
7069impancom 456 . . . . . . 7 ((𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝜑𝑥𝐷) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
7170com12 32 . . . . . 6 ((𝜑𝑥𝐷) → ((𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
7250, 71impbid 202 . . . . 5 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
7372eubidv 2489 . . . 4 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
7473biimpd 219 . . 3 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
75 df-reu 2915 . . 3 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
76 df-reu 2915 . . 3 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
7774, 75, 763imtr4g 285 . 2 ((𝜑𝑥𝐷) → (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
7831, 77mpd 15 1 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃!weu 2469  wne 2790  wnel 2893  ∃!wreu 2910  cdif 3557  wss 3560  {csn 4155  {cpr 4157  cmpt 4683  cfv 5857  crio 6575  (class class class)co 6615  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971   NeighbVtx cnbgr 26145   FriendGraph cfrgr 27020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-edg 25874  df-upgr 25907  df-umgr 25908  df-usgr 25973  df-nbgr 26149  df-frgr 27021
This theorem is referenced by:  frgrncvvdeqlem4  27064  frgrncvvdeqlem5  27065
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