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Mirrors > Home > MPE Home > Th. List > yoneda | Structured version Visualization version GIF version |
Description: The Yoneda Lemma. There is a natural isomorphism between the functors 𝑍 and 𝐸, where 𝑍(𝐹, 𝑋) is the natural transformations from Yon(𝑋) = Hom ( − , 𝑋) to 𝐹, and 𝐸(𝐹, 𝑋) = 𝐹(𝑋) is the evaluation functor. Here we need two universes to state the claim: the smaller universe 𝑈 is used for forming the functor category 𝑄 = 𝐶 op → SetCat(𝑈), which itself does not (necessarily) live in 𝑈 but instead is an element of the larger universe 𝑉. (If 𝑈 is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set 𝑈 = 𝑉 in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.) |
Ref | Expression |
---|---|
yoneda.y | ⊢ 𝑌 = (Yon‘𝐶) |
yoneda.b | ⊢ 𝐵 = (Base‘𝐶) |
yoneda.1 | ⊢ 1 = (Id‘𝐶) |
yoneda.o | ⊢ 𝑂 = (oppCat‘𝐶) |
yoneda.s | ⊢ 𝑆 = (SetCat‘𝑈) |
yoneda.t | ⊢ 𝑇 = (SetCat‘𝑉) |
yoneda.q | ⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
yoneda.h | ⊢ 𝐻 = (HomF‘𝑄) |
yoneda.r | ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
yoneda.e | ⊢ 𝐸 = (𝑂 evalF 𝑆) |
yoneda.z | ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) |
yoneda.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yoneda.w | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
yoneda.u | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
yoneda.v | ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
yoneda.m | ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) |
yoneda.i | ⊢ 𝐼 = (Iso‘𝑅) |
Ref | Expression |
---|---|
yoneda | ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yoneda.r | . . 3 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) | |
2 | 1 | fucbas 18004 | . 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅) |
3 | eqid 2736 | . 2 ⊢ (Inv‘𝑅) = (Inv‘𝑅) | |
4 | yoneda.y | . . . . . . 7 ⊢ 𝑌 = (Yon‘𝐶) | |
5 | yoneda.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
6 | yoneda.1 | . . . . . . 7 ⊢ 1 = (Id‘𝐶) | |
7 | yoneda.o | . . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶) | |
8 | yoneda.s | . . . . . . 7 ⊢ 𝑆 = (SetCat‘𝑈) | |
9 | yoneda.t | . . . . . . 7 ⊢ 𝑇 = (SetCat‘𝑉) | |
10 | yoneda.q | . . . . . . 7 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
11 | yoneda.h | . . . . . . 7 ⊢ 𝐻 = (HomF‘𝑄) | |
12 | yoneda.e | . . . . . . 7 ⊢ 𝐸 = (𝑂 evalF 𝑆) | |
13 | yoneda.z | . . . . . . 7 ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) | |
14 | yoneda.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
15 | yoneda.w | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
16 | yoneda.u | . . . . . . 7 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
17 | yoneda.v | . . . . . . 7 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) | |
18 | 4, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17 | yonedalem1 18313 | . . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
19 | 18 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
20 | funcrcl 17904 | . . . . 5 ⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat)) |
22 | 21 | simpld 494 | . . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
23 | 21 | simprd 495 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Cat) |
24 | 1, 22, 23 | fuccat 18014 | . 2 ⊢ (𝜑 → 𝑅 ∈ Cat) |
25 | 18 | simprd 495 | . 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
26 | yoneda.i | . 2 ⊢ 𝐼 = (Iso‘𝑅) | |
27 | yoneda.m | . . 3 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) | |
28 | eqid 2736 | . . 3 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) | |
29 | 4, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 27, 3, 28 | yonedainv 18322 | . 2 ⊢ (𝜑 → 𝑀(𝑍(Inv‘𝑅)𝐸)(𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))) |
30 | 2, 3, 24, 19, 25, 26, 29 | inviso1 17806 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3948 ⊆ wss 3950 〈cop 4630 ↦ cmpt 5223 ran crn 5684 ‘cfv 6559 (class class class)co 7429 ∈ cmpo 7431 1st c1st 8008 2nd c2nd 8009 tpos ctpos 8246 Basecbs 17243 Hom chom 17304 Catccat 17703 Idccid 17704 Homf chomf 17705 oppCatcoppc 17750 Invcinv 17785 Isociso 17786 Func cfunc 17895 ∘func ccofu 17897 Nat cnat 17985 FuncCat cfuc 17986 SetCatcsetc 18116 ×c cxpc 18209 1stF c1stf 18210 2ndF c2ndf 18211 〈,〉F cprf 18212 evalF cevlf 18250 HomFchof 18289 Yoncyon 18290 |
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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-hom 17317 df-cco 17318 df-cat 17707 df-cid 17708 df-homf 17709 df-comf 17710 df-oppc 17751 df-sect 17787 df-inv 17788 df-iso 17789 df-ssc 17850 df-resc 17851 df-subc 17852 df-func 17899 df-cofu 17901 df-nat 17987 df-fuc 17988 df-setc 18117 df-xpc 18213 df-1stf 18214 df-2ndf 18215 df-prf 18216 df-evlf 18254 df-curf 18255 df-hof 18291 df-yon 18292 |
This theorem is referenced by: (None) |
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