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Theorem frgrncvvdeqlem7 27033
 Description: Lemma 7 for frgrncvvdeq 27038. (Contributed by Alexander van der Vekens, 23-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
frgrncvvdeqlem7 ((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 frgrncvvdeqlem7
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, 10frgrncvvdeqlem6 27032 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6158 . . . 4 (𝐴𝑥) ∈ V
1312snid 4179 . . 3 (𝐴𝑥) ∈ {(𝐴𝑥)}
14 eleq2 2687 . . . . 5 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} ↔ (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
1514biimpd 219 . . . 4 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} → (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
16 elin 3774 . . . . 5 ((𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ↔ ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
17 frgrusgr 26990 . . . . . . . . 9 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
182nbusgreledg 26136 . . . . . . . . . . 11 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ 𝐸))
19 prcom 4237 . . . . . . . . . . . 12 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
2019eleq1i 2689 . . . . . . . . . . 11 ({(𝐴𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸)
2118, 20syl6bb 276 . . . . . . . . . 10 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴𝑥)} ∈ 𝐸))
2221biimpd 219 . . . . . . . . 9 (𝐺 ∈ USGraph → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
239, 17, 223syl 18 . . . . . . . 8 (𝜑 → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2423adantr 481 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2524com12 32 . . . . . 6 ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2625adantr 481 . . . . 5 (((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2716, 26sylbi 207 . . . 4 ((𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
2815, 27syl6com 37 . . 3 ((𝐴𝑥) ∈ {(𝐴𝑥)} → ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)))
2913, 28ax-mp 5 . 2 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸))
3011, 29mpcom 38 1 ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   ∉ wnel 2893   ∩ cin 3554  {csn 4148  {cpr 4150   ↦ cmpt 4673  ‘cfv 5847  ℩crio 6564  (class class class)co 6604  Vtxcvtx 25774  Edgcedg 25839   USGraph cusgr 25937   NeighbVtx cnbgr 26111   FriendGraph cfrgr 26986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-edg 25840  df-upgr 25873  df-umgr 25874  df-usgr 25939  df-nbgr 26115  df-frgr 26987 This theorem is referenced by:  frgrncvvdeqlemB  27035
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