Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrncvvdeqlem3 Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlem3 41471
 Description: Lemma 3 for frgrncvvdeq 41480. 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 480 . . 3 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2680 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrusgr 41432 . . . . . . 7 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
6 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
76nbgrisvtx 40581 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑥 ∈ (𝐺 NeighbVtx 𝑋)) → 𝑥𝑉)
87ex 449 . . . . . . 7 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
91, 5, 83syl 18 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
104, 9syl5bi 231 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
1110imp 444 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
12 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1312adantr 480 . . . 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 41469 . . . . 5 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
21 eldif 3550 . . . . . 6 (𝑌 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑌𝑉 ∧ ¬ 𝑌 ∈ {𝑥}))
22 vsnid 4156 . . . . . . . 8 𝑥 ∈ {𝑥}
23 eleq1 2676 . . . . . . . . 9 (𝑌 = 𝑥 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2423eqcoms 2618 . . . . . . . 8 (𝑥 = 𝑌 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2522, 24mpbiri 247 . . . . . . 7 (𝑥 = 𝑌𝑌 ∈ {𝑥})
2625necon3bi 2808 . . . . . 6 𝑌 ∈ {𝑥} → 𝑥𝑌)
2721, 26simplbiim 657 . . . . 5 (𝑌 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑌)
2820, 27syl 17 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
2911, 13, 283jca 1235 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
306, 14frcond1 41438 . . 3 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
312, 29, 30sylc 63 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
32 prex 4836 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
33 prex 4836 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
3432, 33prss 4291 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
35 simpr 476 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) → {𝑦, 𝑌} ∈ 𝐸)
3634, 35sylbir 224 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → {𝑦, 𝑌} ∈ 𝐸)
3736ad2antll 761 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → {𝑦, 𝑌} ∈ 𝐸)
3815a1i 11 . . . . . . . . . . . . 13 (𝜑𝑁 = (𝐺 NeighbVtx 𝑌))
3938eleq2d 2673 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌)))
4014nbusgreledg 40575 . . . . . . . . . . . . 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 267 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4342adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4443adantr 480 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
4537, 44mpbird 246 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → 𝑦𝑁)
46 simpl 472 . . . . . . . . . 10 (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
4734, 46sylbir 224 . . . . . . . . 9 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → {𝑥, 𝑦} ∈ 𝐸)
4847ad2antll 761 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → {𝑥, 𝑦} ∈ 𝐸)
4945, 48jca 553 . . . . . . 7 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
5049ex 449 . . . . . 6 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
5115eleq2i 2680 . . . . . . . . . . . . 13 (𝑦𝑁𝑦 ∈ (𝐺 NeighbVtx 𝑌))
5251, 41syl5bb 271 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ 𝐸))
5352biimpd 218 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 → {𝑦, 𝑌} ∈ 𝐸))
5453adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 → {𝑦, 𝑌} ∈ 𝐸))
5554impcom 445 . . . . . . . . 9 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → {𝑦, 𝑌} ∈ 𝐸)
566nbgrisvtx 40581 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑦𝑉)
5756ex 449 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) → 𝑦𝑉))
581, 5, 573syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) → 𝑦𝑉))
5951, 58syl5bi 231 . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑁𝑦𝑉))
6059adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → (𝑦𝑁𝑦𝑉))
6160impcom 445 . . . . . . . . . . . 12 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → 𝑦𝑉)
6261ad2antlr 759 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦𝑉)
63 simpl 472 . . . . . . . . . . . . 13 (({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → {𝑦, 𝑌} ∈ 𝐸)
64 id 22 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ∈ 𝐸 → {𝑥, 𝑦} ∈ 𝐸)
6563, 64anim12ci 589 . . . . . . . . . . . 12 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸))
6665, 34sylib 207 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
6762, 66jca 553 . . . . . . . . . 10 ((({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
6867ex 449 . . . . . . . . 9 (({𝑦, 𝑌} ∈ 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
6955, 68mpancom 700 . . . . . . . 8 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)))
7069impancom 455 . . . . . . 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 201 . . . . 5 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
7372eubidv 2478 . . . 4 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
7473biimpd 218 . . 3 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
75 df-reu 2903 . . 3 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
76 df-reu 2903 . . 3 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
7774, 75, 763imtr4g 284 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780   ∉ wnel 2781  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlem4  41472  frgrncvvdeqlem5  41473
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