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Mirrors > Home > MPE Home > Th. List > yonffth | Structured version Visualization version GIF version |
Description: The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category πΆ as a full subcategory of the category π of presheaves on πΆ. (Contributed by Mario Carneiro, 29-Jan-2017.) |
Ref | Expression |
---|---|
yonffth.y | β’ π = (YonβπΆ) |
yonffth.o | β’ π = (oppCatβπΆ) |
yonffth.s | β’ π = (SetCatβπ) |
yonffth.q | β’ π = (π FuncCat π) |
yonffth.c | β’ (π β πΆ β Cat) |
yonffth.u | β’ (π β π β π) |
yonffth.h | β’ (π β ran (Homf βπΆ) β π) |
Ref | Expression |
---|---|
yonffth | β’ (π β π β ((πΆ Full π) β© (πΆ Faith π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yonffth.y | . 2 β’ π = (YonβπΆ) | |
2 | eqid 2731 | . 2 β’ (BaseβπΆ) = (BaseβπΆ) | |
3 | eqid 2731 | . 2 β’ (IdβπΆ) = (IdβπΆ) | |
4 | yonffth.o | . 2 β’ π = (oppCatβπΆ) | |
5 | yonffth.s | . 2 β’ π = (SetCatβπ) | |
6 | eqid 2731 | . 2 β’ (SetCatβ(ran (Homf βπ) βͺ π)) = (SetCatβ(ran (Homf βπ) βͺ π)) | |
7 | yonffth.q | . 2 β’ π = (π FuncCat π) | |
8 | eqid 2731 | . 2 β’ (HomFβπ) = (HomFβπ) | |
9 | eqid 2731 | . 2 β’ ((π Γc π) FuncCat (SetCatβ(ran (Homf βπ) βͺ π))) = ((π Γc π) FuncCat (SetCatβ(ran (Homf βπ) βͺ π))) | |
10 | eqid 2731 | . 2 β’ (π evalF π) = (π evalF π) | |
11 | eqid 2731 | . 2 β’ ((HomFβπ) βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) = ((HomFβπ) βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) | |
12 | yonffth.c | . 2 β’ (π β πΆ β Cat) | |
13 | fvex 6904 | . . . 4 β’ (Homf βπ) β V | |
14 | 13 | rnex 7906 | . . 3 β’ ran (Homf βπ) β V |
15 | yonffth.u | . . 3 β’ (π β π β π) | |
16 | unexg 7739 | . . 3 β’ ((ran (Homf βπ) β V β§ π β π) β (ran (Homf βπ) βͺ π) β V) | |
17 | 14, 15, 16 | sylancr 586 | . 2 β’ (π β (ran (Homf βπ) βͺ π) β V) |
18 | yonffth.h | . 2 β’ (π β ran (Homf βπΆ) β π) | |
19 | ssidd 4005 | . 2 β’ (π β (ran (Homf βπ) βͺ π) β (ran (Homf βπ) βͺ π)) | |
20 | eqid 2731 | . 2 β’ (π β (π Func π), π₯ β (BaseβπΆ) β¦ (π β (((1st βπ)βπ₯)(π Nat π)π) β¦ ((πβπ₯)β((IdβπΆ)βπ₯)))) = (π β (π Func π), π₯ β (BaseβπΆ) β¦ (π β (((1st βπ)βπ₯)(π Nat π)π) β¦ ((πβπ₯)β((IdβπΆ)βπ₯)))) | |
21 | eqid 2731 | . 2 β’ (Invβ((π Γc π) FuncCat (SetCatβ(ran (Homf βπ) βͺ π)))) = (Invβ((π Γc π) FuncCat (SetCatβ(ran (Homf βπ) βͺ π)))) | |
22 | eqid 2731 | . 2 β’ (π β (π Func π), π₯ β (BaseβπΆ) β¦ (π’ β ((1st βπ)βπ₯) β¦ (π¦ β (BaseβπΆ) β¦ (π β (π¦(Hom βπΆ)π₯) β¦ (((π₯(2nd βπ)π¦)βπ)βπ’))))) = (π β (π Func π), π₯ β (BaseβπΆ) β¦ (π’ β ((1st βπ)βπ₯) β¦ (π¦ β (BaseβπΆ) β¦ (π β (π¦(Hom βπΆ)π₯) β¦ (((π₯(2nd βπ)π¦)βπ)βπ’))))) | |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22 | yonffthlem 18240 | 1 β’ (π β π β ((πΆ Full π) β© (πΆ Faith π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3473 βͺ cun 3946 β© cin 3947 β wss 3948 β¨cop 4634 β¦ cmpt 5231 ran crn 5677 βcfv 6543 (class class class)co 7412 β cmpo 7414 1st c1st 7976 2nd c2nd 7977 tpos ctpos 8213 Basecbs 17149 Hom chom 17213 Catccat 17613 Idccid 17614 Homf chomf 17615 oppCatcoppc 17660 Invcinv 17697 Func cfunc 17809 βfunc ccofu 17811 Full cful 17858 Faith cfth 17859 Nat cnat 17897 FuncCat cfuc 17898 SetCatcsetc 18030 Γc cxpc 18125 1stF c1stf 18126 2ndF c2ndf 18127 β¨,β©F cprf 18128 evalF cevlf 18167 HomFchof 18206 Yoncyon 18207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-homf 17619 df-comf 17620 df-oppc 17661 df-sect 17699 df-inv 17700 df-iso 17701 df-ssc 17762 df-resc 17763 df-subc 17764 df-func 17813 df-cofu 17815 df-full 17860 df-fth 17861 df-nat 17899 df-fuc 17900 df-setc 18031 df-xpc 18129 df-1stf 18130 df-2ndf 18131 df-prf 18132 df-evlf 18171 df-curf 18172 df-hof 18208 df-yon 18209 |
This theorem is referenced by: yoniso 18243 |
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