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Theorem frgrncvvdeqlem6 27284
 Description: Lemma 6 for frgrncvvdeq 27289. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-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
frgrncvvdeqlem6 ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)
Distinct variable groups:   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   𝑦,𝐸   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem6
StepHypRef Expression
1 frgrncvvdeq.v1 . . 3 𝑉 = (Vtx‘𝐺)
2 frgrncvvdeq.e . . 3 𝐸 = (Edg‘𝐺)
3 frgrncvvdeq.nx . . 3 𝐷 = (𝐺 NeighbVtx 𝑋)
4 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
5 frgrncvvdeq.x . . 3 (𝜑𝑋𝑉)
6 frgrncvvdeq.y . . 3 (𝜑𝑌𝑉)
7 frgrncvvdeq.ne . . 3 (𝜑𝑋𝑌)
8 frgrncvvdeq.xy . . 3 (𝜑𝑌𝐷)
9 frgrncvvdeq.f . . 3 (𝜑𝐺 ∈ FriendGraph )
10 frgrncvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem5 27283 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6239 . . . . 5 (𝐴𝑥) ∈ V
13 elinsn 4277 . . . . 5 (((𝐴𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)}) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
1412, 13mpan 706 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)} → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
15 frgrusgr 27240 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
162nbusgreledg 26294 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ 𝐸))
17 prcom 4299 . . . . . . . . . . 11 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
1817eleq1i 2721 . . . . . . . . . 10 ({(𝐴𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸)
1916, 18syl6bb 276 . . . . . . . . 9 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸))
2019biimpd 219 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
219, 15, 203syl 18 . . . . . . 7 (𝜑 → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2221adantr 480 . . . . . 6 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2322com12 32 . . . . 5 ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2423adantr 480 . . . 4 (((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2514, 24syl 17 . . 3 (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)} → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2625eqcoms 2659 . 2 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2711, 26mpcom 38 1 ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ∉ wnel 2926  Vcvv 3231   ∩ cin 3606  {csn 4210  {cpr 4212   ↦ cmpt 4762  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269   FriendGraph cfrgr 27236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-nbgr 26270  df-frgr 27237 This theorem is referenced by:  frgrncvvdeqlem8  27286
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