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Mirrors > Home > MPE Home > Th. List > uhgrstrrepe | Structured version Visualization version GIF version |
Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
uhgrstrrepe.v | ⊢ 𝑉 = (Base‘𝐺) |
uhgrstrrepe.i | ⊢ 𝐼 = (.ef‘ndx) |
uhgrstrrepe.s | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
uhgrstrrepe.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) |
uhgrstrrepe.w | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
uhgrstrrepe.e | ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
Ref | Expression |
---|---|
uhgrstrrepe | ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrstrrepe.e | . . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) | |
2 | uhgrstrrepe.i | . . . . . . . . 9 ⊢ 𝐼 = (.ef‘ndx) | |
3 | uhgrstrrepe.s | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
4 | uhgrstrrepe.b | . . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
5 | uhgrstrrepe.w | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
6 | 2, 3, 4, 5 | setsvtx 26383 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | uhgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | syl6eqr 2831 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4383 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | difeq1d 3949 | . . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 10 | feq3d 6278 | . . . 4 ⊢ (𝜑 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
12 | 1, 11 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅})) |
13 | 2, 3, 4, 5 | setsiedg 26384 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
14 | 13 | dmeqd 5571 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
15 | 13, 14 | feq12d 6279 | . . 3 ⊢ (𝜑 → ((iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
16 | 12, 15 | mpbird 249 | . 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅})) |
17 | ovex 6954 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
18 | eqid 2777 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
19 | eqid 2777 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
20 | 18, 19 | isuhgr 26408 | . . 3 ⊢ ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ V → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
21 | 17, 20 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
22 | 16, 21 | mpbird 249 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∖ cdif 3788 ∅c0 4140 𝒫 cpw 4378 {csn 4397 〈cop 4403 class class class wbr 4886 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Struct cstr 16251 ndxcnx 16252 sSet csts 16253 Basecbs 16255 .efcedgf 26337 Vtxcvtx 26344 iEdgciedg 26345 UHGraphcuhgr 26404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-edgf 26338 df-vtx 26346 df-iedg 26347 df-uhgr 26406 |
This theorem is referenced by: (None) |
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