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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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