Theorem frgrncvvdeqlem6 27841

Description: Lemma 6 for frgrncvvdeq 27846. (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,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   𝑦,𝐸   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 27840	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6514	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13		elinsn 4521	. . . . 5 ⊢ (((𝐴‘𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)}) → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
14	12, 13	mpan 677	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)} → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
15		frgrusgr 27797	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
16	2	nbusgreledg 26841	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴‘𝑥), 𝑥} ∈ 𝐸))
17		prcom 4543	. . . . . . . . . . 11 ⊢ {(𝐴‘𝑥), 𝑥} = {𝑥, (𝐴‘𝑥)}
18	17	eleq1i 2856	. . . . . . . . . 10 ⊢ ({(𝐴‘𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸)
19	16, 18	syl6bb 279	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
20	19	biimpd 221	. . . . . . . 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 473	. . . . . 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 473	. . . 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 2786	. 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 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2967   ∉ wnel 3073  Vcvv 3415   ∩ cin 3830  {csn 4442  {cpr 4444   ↦ cmpt 5009  ‘cfv 6190  ℩crio 6938  (class class class)co 6978  Vtxcvtx 26487  Edgcedg 26538  USGraphcusgr 26640   NeighbVtx cnbgr 26820   FriendGraph cfrgr 27793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-2o 7908  df-oadd 7911  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-dju 9126  df-card 9164  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-2 11506  df-n0 11711  df-xnn0 11783  df-z 11797  df-uz 12062  df-fz 12712  df-hash 13509  df-edg 26539  df-upgr 26573  df-umgr 26574  df-usgr 26642  df-nbgr 26821  df-frgr 27794

This theorem is referenced by:  frgrncvvdeqlem8  27843