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Theorem frgrncvvdeqlem6 27148
Description: Lemma 6 for frgrncvvdeq 27153. (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,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦,𝑥   𝑦,𝐸   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,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 27147 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6188 . . . . 5 (𝐴𝑥) ∈ V
13 elinsn 4236 . . . . 5 (((𝐴𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)}) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
1412, 13mpan 705 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)} → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
15 frgrusgr 27104 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
162nbusgreledg 26230 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ 𝐸))
17 prcom 4258 . . . . . . . . . . 11 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
1817eleq1i 2690 . . . . . . . . . 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 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 2628 . 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 384   = wceq 1481  wcel 1988  wne 2791  wnel 2894  Vcvv 3195  cin 3566  {csn 4168  {cpr 4170  cmpt 4720  cfv 5876  crio 6595  (class class class)co 6635  Vtxcvtx 25855  Edgcedg 25920   USGraph cusgr 26025   NeighbVtx cnbgr 26205   FriendGraph cfrgr 27100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-edg 25921  df-upgr 25958  df-umgr 25959  df-usgr 26027  df-nbgr 26209  df-frgr 27101
This theorem is referenced by:  frgrncvvdeqlem8  27150
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