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Theorem frgrncvvdeqlem6 29546
Description: Lemma 6 for frgrncvvdeq 29551. (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,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,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 29545 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6901 . . . . 5 (𝐴𝑥) ∈ V
13 elinsn 4713 . . . . 5 (((𝐴𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)}) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
1412, 13mpan 688 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)} → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
15 frgrusgr 29503 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
162nbusgreledg 28599 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ 𝐸))
17 prcom 4735 . . . . . . . . . . 11 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
1817eleq1i 2824 . . . . . . . . . 10 ({(𝐴𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸)
1916, 18bitrdi 286 . . . . . . . . 9 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸))
2019biimpd 228 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
219, 15, 203syl 18 . . . . . . 7 (𝜑 → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2221adantr 481 . . . . . 6 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2322com12 32 . . . . 5 ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2423adantr 481 . . . 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 2740 . 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 396   = wceq 1541  wcel 2106  wne 2940  wnel 3046  Vcvv 3474  cin 3946  {csn 4627  {cpr 4629  cmpt 5230  cfv 6540  crio 7360  (class class class)co 7405  Vtxcvtx 28245  Edgcedg 28296  USGraphcusgr 28398   NeighbVtx cnbgr 28578   FriendGraph cfrgr 29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-edg 28297  df-upgr 28331  df-umgr 28332  df-usgr 28400  df-nbgr 28579  df-frgr 29501
This theorem is referenced by:  frgrncvvdeqlem8  29548
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