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Mirrors > Home > MPE Home > Th. List > cusgrexg | Structured version Visualization version GIF version |
Description: For each set there is a set of edges so that the set together with these edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
Ref | Expression |
---|---|
cusgrexg | ⊢ (𝑉 ∈ 𝑊 → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6776 | . . . 4 ⊢ (𝑦 = 𝑥 → ((♯‘𝑦) = 2 ↔ (♯‘𝑥) = 2)) | |
2 | 1 | cbvrabv 3424 | . . 3 ⊢ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
3 | 2 | cusgrexilem1 27794 | . 2 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2}) ∈ V) |
4 | 2 | cusgrexi 27798 | . 2 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2})〉 ∈ ComplUSGraph) |
5 | opeq2 4806 | . . 3 ⊢ (𝑒 = ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2}) → 〈𝑉, 𝑒〉 = 〈𝑉, ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2})〉) | |
6 | 5 | eleq1d 2823 | . 2 ⊢ (𝑒 = ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2}) → (〈𝑉, 𝑒〉 ∈ ComplUSGraph ↔ 〈𝑉, ( I ↾ {𝑦 ∈ 𝒫 𝑉 ∣ (♯‘𝑦) = 2})〉 ∈ ComplUSGraph)) |
7 | 3, 4, 6 | spcedv 3535 | 1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {crab 3068 Vcvv 3430 𝒫 cpw 4534 〈cop 4568 I cid 5484 ↾ cres 5587 ‘cfv 6427 2c2 12016 ♯chash 14032 ComplUSGraphccusgr 27765 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-oadd 8289 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-dju 9647 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-hash 14033 df-vtx 27356 df-iedg 27357 df-edg 27406 df-usgr 27509 df-nbgr 27688 df-uvtx 27741 df-cplgr 27766 df-cusgr 27767 |
This theorem is referenced by: fusgrmaxsize 27819 |
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