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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for frgrncvvdeq 28416. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-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 |
---|---|
frgrncvvdeqlem5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
2 | riotaex 7193 | . . . 4 ⊢ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ V | |
3 | frgrncvvdeq.a | . . . . 5 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | |
4 | 3 | fvmpt2 6848 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ V) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
5 | 1, 2, 4 | sylancl 589 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
6 | 5 | sneqd 4568 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}) |
7 | frgrncvvdeq.v1 | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | frgrncvvdeq.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
9 | frgrncvvdeq.nx | . . 3 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
10 | frgrncvvdeq.ny | . . 3 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
11 | frgrncvvdeq.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
12 | frgrncvvdeq.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | frgrncvvdeq.ne | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
14 | frgrncvvdeq.xy | . . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
15 | frgrncvvdeq.f | . . 3 ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) | |
16 | 7, 8, 9, 10, 11, 12, 13, 14, 15, 3 | frgrncvvdeqlem3 28408 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
17 | 6, 16 | eqtrd 2778 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ∉ wnel 3047 Vcvv 3421 ∩ cin 3880 {csn 4556 {cpr 4558 ↦ cmpt 5150 ‘cfv 6398 ℩crio 7188 (class class class)co 7232 Vtxcvtx 27111 Edgcedg 27162 NeighbVtx cnbgr 27444 FriendGraph cfrgr 28365 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-2o 8224 df-oadd 8227 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-dju 9542 df-card 9580 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-n0 12116 df-xnn0 12188 df-z 12202 df-uz 12464 df-fz 13121 df-hash 13922 df-edg 27163 df-upgr 27197 df-umgr 27198 df-usgr 27266 df-nbgr 27445 df-frgr 28366 |
This theorem is referenced by: frgrncvvdeqlem6 28411 frgrncvvdeqlem7 28412 frgrncvvdeqlem9 28414 |
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