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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for frgrncvvdeq 29562. 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 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
2 | riotaex 7369 | . . . 4 ⊢ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ V | |
3 | frgrncvvdeq.a | . . . . 5 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | |
4 | 3 | fvmpt2 7010 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ V) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
5 | 1, 2, 4 | sylancl 587 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
6 | 5 | sneqd 4641 | . 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 29554 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
17 | 6, 16 | eqtrd 2773 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 Vcvv 3475 ∩ cin 3948 {csn 4629 {cpr 4631 ↦ cmpt 5232 ‘cfv 6544 ℩crio 7364 (class class class)co 7409 Vtxcvtx 28256 Edgcedg 28307 NeighbVtx cnbgr 28589 FriendGraph cfrgr 29511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-edg 28308 df-upgr 28342 df-umgr 28343 df-usgr 28411 df-nbgr 28590 df-frgr 29512 |
This theorem is referenced by: frgrncvvdeqlem6 29557 frgrncvvdeqlem7 29558 frgrncvvdeqlem9 29560 |
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