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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem6 | Structured version Visualization version GIF version |
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.) |
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 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) |
Colors of variables: wff setvar class |
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 |
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