| 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 47929 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑉) |
| 4 | eqid 2737 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
| 5 | 2, 4 | dfnbgr2 29354 | . . 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 2830 | . . . . 5 ⊢ (𝑒 = {0, 𝑥} → (0 ∈ 𝑒 ↔ 0 ∈ {0, 𝑥})) | |
| 8 | eleq2 2830 | . . . . 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 13664 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 0 ∈ (0...𝑁)) |
| 12 | fz1ssfz0 13663 | . . . . . . 7 ⊢ (1...𝑁) ⊆ (0...𝑁) | |
| 13 | 1 | fveq2i 6909 | . . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘(StarGr‘𝑁)) |
| 14 | 2, 13 | eqtri 2765 | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘(StarGr‘𝑁)) |
| 15 | stgrvtx 47921 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁)) | |
| 16 | 14, 15 | eqtrid 2789 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑉 = (0...𝑁)) |
| 17 | 16 | difeq1d 4125 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑉 ∖ {0}) = ((0...𝑁) ∖ {0})) |
| 18 | fz0dif1 13646 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → ((0...𝑁) ∖ {0}) = (1...𝑁)) | |
| 19 | 18 | eqimssd 4040 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ((0...𝑁) ∖ {0}) ⊆ (1...𝑁)) |
| 20 | 17, 19 | eqsstrd 4018 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑉 ∖ {0}) ⊆ (1...𝑁)) |
| 21 | 20 | sselda 3983 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ (1...𝑁)) |
| 22 | 12, 21 | sselid 3981 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ (0...𝑁)) |
| 23 | 11, 22 | prssd 4822 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → {0, 𝑥} ⊆ (0...𝑁)) |
| 24 | preq2 4734 | . . . . . . 7 ⊢ (𝑛 = 𝑥 → {0, 𝑛} = {0, 𝑥}) | |
| 25 | 24 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑛 = 𝑥 → ({0, 𝑥} = {0, 𝑛} ↔ {0, 𝑥} = {0, 𝑥})) |
| 26 | eqidd 2738 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → {0, 𝑥} = {0, 𝑥}) | |
| 27 | 25, 21, 26 | rspcedvdw 3625 | . . . . 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 2833 | . . . . . 6 ⊢ ({0, 𝑥} ∈ (Edg‘𝐺) ↔ {0, 𝑥} ∈ (Edg‘(StarGr‘𝑁))) |
| 30 | stgredgel 47924 | . . . . . . 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 4761 | . . . . . 6 ⊢ (𝑥 ∈ (𝑉 ∖ {0}) → 𝑥 ∈ {0, 𝑥}) | |
| 35 | 34 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → 𝑥 ∈ {0, 𝑥}) |
| 36 | c0ex 11255 | . . . . . 6 ⊢ 0 ∈ V | |
| 37 | 36 | prid1 4762 | . . . . 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 3625 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ (𝑉 ∖ {0})) → ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)) |
| 40 | 39 | rabeqcda 3448 | . 2 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ (𝑉 ∖ {0}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒)} = (𝑉 ∖ {0})) |
| 41 | 6, 40 | eqtrd 2777 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = (𝑉 ∖ {0})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℕ0cn0 12526 ...cfz 13547 Vtxcvtx 29013 Edgcedg 29064 NeighbVtx cnbgr 29349 StarGrcstgr 47918 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-hash 14370 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-edgf 29004 df-vtx 29015 df-iedg 29016 df-edg 29065 df-nbgr 29350 df-stgr 47919 |
| This theorem is referenced by: stgrclnbgr0 47932 |
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