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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stgrnbgr0 | Structured version Visualization version GIF version |
Description: All vertices of a star graph SN except the center are in the (open) neighborhood of the center. (Contributed by AV, 12-Sep-2025.) |
Ref | Expression |
---|---|
stgrvtx0.g | ⊢ 𝐺 = (StarGr‘𝑁) |
stgrvtx0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
stgrnbgr0 | ⊢ (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = (𝑉 ∖ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stgrvtx0.g | . . . 4 ⊢ 𝐺 = (StarGr‘𝑁) | |
2 | stgrvtx0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | stgrvtx0 47864 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑉) |
4 | eqid 2734 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
5 | 2, 4 | dfnbgr2 29368 | . . 3 ⊢ (0 ∈ 𝑉 → (𝐺 NeighbVtx 0) = {𝑥 ∈ (𝑉 ∖ {0}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)}) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = {𝑥 ∈ (𝑉 ∖ {0}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)}) |
7 | eleq2 2827 | . . . . 5 ⊢ (𝑒 = {0, 𝑥} → (0 ∈ 𝑒 ↔ 0 ∈ {0, 𝑥})) | |
8 | eleq2 2827 | . . . . 5 ⊢ (𝑒 = {0, 𝑥} → (𝑥 ∈ 𝑒 ↔ 𝑥 ∈ {0, 𝑥})) | |
9 | 7, 8 | anbi12d 632 | . . . 4 ⊢ (𝑒 = {0, 𝑥} → ((0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒) ↔ (0 ∈ {0, 𝑥} ∧ 𝑥 ∈ {0, 𝑥}))) |
10 | 0elfz 13660 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 0 ∈ (0...𝑁)) |
12 | fz1ssfz0 13659 | . . . . . . 7 ⊢ (1...𝑁) ⊆ (0...𝑁) | |
13 | 1 | fveq2i 6909 | . . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘(StarGr‘𝑁)) |
14 | 2, 13 | eqtri 2762 | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘(StarGr‘𝑁)) |
15 | stgrvtx 47856 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁)) | |
16 | 14, 15 | eqtrid 2786 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑉 = (0...𝑁)) |
17 | 16 | difeq1d 4134 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑉 ∖ {0}) = ((0...𝑁) ∖ {0})) |
18 | fz0dif1 13642 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → ((0...𝑁) ∖ {0}) = (1...𝑁)) | |
19 | 18 | eqimssd 4051 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ((0...𝑁) ∖ {0}) ⊆ (1...𝑁)) |
20 | 17, 19 | eqsstrd 4033 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑉 ∖ {0}) ⊆ (1...𝑁)) |
21 | 20 | sselda 3994 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ (1...𝑁)) |
22 | 12, 21 | sselid 3992 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ (0...𝑁)) |
23 | 11, 22 | prssd 4826 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → {0, 𝑥} ⊆ (0...𝑁)) |
24 | preq2 4738 | . . . . . . 7 ⊢ (𝑛 = 𝑥 → {0, 𝑛} = {0, 𝑥}) | |
25 | 24 | eqeq2d 2745 | . . . . . 6 ⊢ (𝑛 = 𝑥 → ({0, 𝑥} = {0, 𝑛} ↔ {0, 𝑥} = {0, 𝑥})) |
26 | eqidd 2735 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → {0, 𝑥} = {0, 𝑥}) | |
27 | 25, 21, 26 | rspcedvdw 3624 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → ∃𝑛 ∈ (1...𝑁){0, 𝑥} = {0, 𝑛}) |
28 | 1 | fveq2i 6909 | . . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘(StarGr‘𝑁)) |
29 | 28 | eleq2i 2830 | . . . . . 6 ⊢ ({0, 𝑥} ∈ (Edg‘𝐺) ↔ {0, 𝑥} ∈ (Edg‘(StarGr‘𝑁))) |
30 | stgredgel 47859 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ({0, 𝑥} ∈ (Edg‘(StarGr‘𝑁)) ↔ ({0, 𝑥} ⊆ (0...𝑁) ∧ ∃𝑛 ∈ (1...𝑁){0, 𝑥} = {0, 𝑛}))) | |
31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → ({0, 𝑥} ∈ (Edg‘(StarGr‘𝑁)) ↔ ({0, 𝑥} ⊆ (0...𝑁) ∧ ∃𝑛 ∈ (1...𝑁){0, 𝑥} = {0, 𝑛}))) |
32 | 29, 31 | bitrid 283 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → ({0, 𝑥} ∈ (Edg‘𝐺) ↔ ({0, 𝑥} ⊆ (0...𝑁) ∧ ∃𝑛 ∈ (1...𝑁){0, 𝑥} = {0, 𝑛}))) |
33 | 23, 27, 32 | mpbir2and 713 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → {0, 𝑥} ∈ (Edg‘𝐺)) |
34 | prid2g 4765 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉 ∖ {0}) → 𝑥 ∈ {0, 𝑥}) | |
35 | 34 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ {0, 𝑥}) |
36 | c0ex 11252 | . . . . . 6 ⊢ 0 ∈ V | |
37 | 36 | prid1 4766 | . . . . 5 ⊢ 0 ∈ {0, 𝑥} |
38 | 35, 37 | jctil 519 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → (0 ∈ {0, 𝑥} ∧ 𝑥 ∈ {0, 𝑥})) |
39 | 9, 33, 38 | rspcedvdw 3624 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)) |
40 | 39 | rabeqcda 3444 | . 2 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ (𝑉 ∖ {0}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)} = (𝑉 ∖ {0})) |
41 | 6, 40 | eqtrd 2774 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = (𝑉 ∖ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {crab 3432 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 {cpr 4632 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 ℕ0cn0 12523 ...cfz 13543 Vtxcvtx 29027 Edgcedg 29078 NeighbVtx cnbgr 29363 StarGrcstgr 47853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-hash 14366 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-edgf 29018 df-vtx 29029 df-iedg 29030 df-edg 29079 df-nbgr 29364 df-stgr 47854 |
This theorem is referenced by: stgrclnbgr0 47867 |
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