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Theorem frgrncvvdeqlem6 28089
Description: Lemma 6 for frgrncvvdeq 28094. (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 28088 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6658 . . . . 5 (𝐴𝑥) ∈ V
13 elinsn 4606 . . . . 5 (((𝐴𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)}) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
1412, 13mpan 689 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴𝑥)} → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
15 frgrusgr 28046 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
162nbusgreledg 27143 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ 𝐸))
17 prcom 4628 . . . . . . . . . . 11 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
1817eleq1i 2880 . . . . . . . . . 10 ({(𝐴𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸)
1916, 18syl6bb 290 . . . . . . . . 9 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸))
2019biimpd 232 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
219, 15, 203syl 18 . . . . . . 7 (𝜑 → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2221adantr 484 . . . . . 6 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2322com12 32 . . . . 5 ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2423adantr 484 . . . 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 2806 . 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 399   = wceq 1538  wcel 2111  wne 2987  wnel 3091  Vcvv 3441  cin 3880  {csn 4525  {cpr 4527  cmpt 5110  cfv 6324  crio 7092  (class class class)co 7135  Vtxcvtx 26789  Edgcedg 26840  USGraphcusgr 26942   NeighbVtx cnbgr 27122   FriendGraph cfrgr 28043
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-edg 26841  df-upgr 26875  df-umgr 26876  df-usgr 26944  df-nbgr 27123  df-frgr 28044
This theorem is referenced by:  frgrncvvdeqlem8  28091
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