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Mirrors > Home > MPE Home > Th. List > Mathboxes > oduoppcbas | Structured version Visualization version GIF version |
Description: The dual of a preordered set and the opposite category have the same set of objects. (Contributed by Zhi Wang, 22-Sep-2025.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
oduoppcbas.d | ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾))) |
oduoppcbas.o | ⊢ 𝑂 = (oppCat‘𝐶) |
Ref | Expression |
---|---|
oduoppcbas | ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prstcnid.c | . . 3 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
2 | prstcnid.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
3 | oduoppcbas.d | . . . . 5 ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾))) | |
4 | eqid 2734 | . . . . . . 7 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
5 | 4 | oduprs 18357 | . . . . . 6 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset ) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (ODual‘𝐾) ∈ Proset ) |
7 | eqid 2734 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 4, 7 | odubas 18347 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘(ODual‘𝐾)) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(ODual‘𝐾))) |
10 | 3, 6, 9 | prstcbas 48867 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐷)) |
11 | 10 | eqcomd 2740 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐾)) |
12 | 1, 2, 11 | prstcbas 48867 | . 2 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐶)) |
13 | oduoppcbas.o | . . 3 ⊢ 𝑂 = (oppCat‘𝐶) | |
14 | eqid 2734 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
15 | 13, 14 | oppcbas 17763 | . 2 ⊢ (Base‘𝐶) = (Base‘𝑂) |
16 | 12, 15 | eqtrdi 2790 | 1 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 Basecbs 17244 oppCatcoppc 17755 ODualcodu 18342 Proset cproset 18349 ProsetToCatcprstc 48862 |
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-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-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-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 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-z 12611 df-dec 12731 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ple 17317 df-hom 17321 df-cco 17322 df-oppc 17756 df-odu 18343 df-proset 18351 df-prstc 48863 |
This theorem is referenced by: oduoppcciso 48881 |
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