Theorem frgrncvvdeqlem6 28083

Description: Lemma 6 for frgrncvvdeq 28088. (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	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 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

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		frgrncvvdeq.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		frgrncvvdeq.ne	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
8		frgrncvvdeq.xy	. . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
9		frgrncvvdeq.f	. . 3 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
10		frgrncvvdeq.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem5 28082	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6683	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13		elinsn 4646	. . . . 5 ⊢ (((𝐴‘𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)}) → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
14	12, 13	mpan 688	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)} → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
15		frgrusgr 28040	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
16	2	nbusgreledg 27135	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴‘𝑥), 𝑥} ∈ 𝐸))
17		prcom 4668	. . . . . . . . . . 11 ⊢ {(𝐴‘𝑥), 𝑥} = {𝑥, (𝐴‘𝑥)}
18	17	eleq1i 2903	. . . . . . . . . 10 ⊢ ({(𝐴‘𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸)
19	16, 18	syl6bb 289	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
20	19	biimpd 231	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
21	9, 15, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
22	21	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
23	22	com12 32	. . . . 5 ⊢ ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
24	23	adantr 483	. . . 4 ⊢ (((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
25	14, 24	syl 17	. . 3 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)} → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
26	25	eqcoms 2829	. 2 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
27	11, 26	mpcom 38	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∉ wnel 3123  Vcvv 3494   ∩ cin 3935  {csn 4567  {cpr 4569   ↦ cmpt 5146  ‘cfv 6355  ℩crio 7113  (class class class)co 7156  Vtxcvtx 26781  Edgcedg 26832  USGraphcusgr 26934   NeighbVtx cnbgr 27114   FriendGraph cfrgr 28037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-edg 26833  df-upgr 26867  df-umgr 26868  df-usgr 26936  df-nbgr 27115  df-frgr 28038

This theorem is referenced by:  frgrncvvdeqlem8  28085