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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 17078 | . 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅) |
3 | eqid 2772 | . 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 17370 | . . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
19 | 18 | simpld 487 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
20 | funcrcl 16981 | . . . . 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 487 | . . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
23 | 21 | simprd 488 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Cat) |
24 | 1, 22, 23 | fuccat 17088 | . 2 ⊢ (𝜑 → 𝑅 ∈ Cat) |
25 | 18 | simprd 488 | . 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 2772 | . . 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 17379 | . 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 16884 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∪ cun 3823 ⊆ wss 3825 〈cop 4441 ↦ cmpt 5002 ran crn 5401 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 1st c1st 7492 2nd c2nd 7493 tpos ctpos 7687 Basecbs 16329 Hom chom 16422 Catccat 16783 Idccid 16784 Homf chomf 16785 oppCatcoppc 16829 Invcinv 16863 Isociso 16864 Func cfunc 16972 ∘func ccofu 16974 Nat cnat 17059 FuncCat cfuc 17060 SetCatcsetc 17183 ×c cxpc 17266 1stF c1stf 17267 2ndF c2ndf 17268 〈,〉F cprf 17269 evalF cevlf 17307 HomFchof 17346 Yoncyon 17347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-hom 16435 df-cco 16436 df-cat 16787 df-cid 16788 df-homf 16789 df-comf 16790 df-oppc 16830 df-sect 16865 df-inv 16866 df-iso 16867 df-ssc 16928 df-resc 16929 df-subc 16930 df-func 16976 df-cofu 16978 df-nat 17061 df-fuc 17062 df-setc 17184 df-xpc 17270 df-1stf 17271 df-2ndf 17272 df-prf 17273 df-evlf 17311 df-curf 17312 df-hof 17348 df-yon 17349 |
This theorem is referenced by: (None) |
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