Theorem frgrncvvdeqlem6 28359

Description: Lemma 6 for frgrncvvdeq 28364. (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 28358	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6719	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13		elinsn 4616	. . . . 5 ⊢ (((𝐴‘𝑥) ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)}) → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
14	12, 13	mpan 690	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = {(𝐴‘𝑥)} → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
15		frgrusgr 28316	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
16	2	nbusgreledg 27413	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {(𝐴‘𝑥), 𝑥} ∈ 𝐸))
17		prcom 4638	. . . . . . . . . . 11 ⊢ {(𝐴‘𝑥), 𝑥} = {𝑥, (𝐴‘𝑥)}
18	17	eleq1i 2824	. . . . . . . . . 10 ⊢ ({(𝐴‘𝑥), 𝑥} ∈ 𝐸 ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸)
19	16, 18	bitrdi 290	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, (𝐴‘𝑥)} ∈ 𝐸))
20	19	biimpd 232	. . . . . . . 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 484	. . . . . 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 484	. . . 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 2742	. 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 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2935   ∉ wnel 3039  Vcvv 3401   ∩ cin 3856  {csn 4531  {cpr 4533   ↦ cmpt 5124  ‘cfv 6369  ℩crio 7158  (class class class)co 7202  Vtxcvtx 27059  Edgcedg 27110  USGraphcusgr 27212   NeighbVtx cnbgr 27392   FriendGraph cfrgr 28313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-oadd 8195  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-dju 9500  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-n0 12074  df-xnn0 12146  df-z 12160  df-uz 12422  df-fz 13079  df-hash 13880  df-edg 27111  df-upgr 27145  df-umgr 27146  df-usgr 27214  df-nbgr 27393  df-frgr 28314

This theorem is referenced by:  frgrncvvdeqlem8  28361