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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem6 | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
frgrncvvdeqlem6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸) |
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) |
Colors of variables: wff setvar class |
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 |
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