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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 17468 | . 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅) |
3 | eqid 2737 | . 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 17780 | . . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
19 | 18 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
20 | funcrcl 17369 | . . . . 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 498 | . . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
23 | 21 | simprd 499 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Cat) |
24 | 1, 22, 23 | fuccat 17479 | . 2 ⊢ (𝜑 → 𝑅 ∈ Cat) |
25 | 18 | simprd 499 | . 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 2737 | . . 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 17789 | . 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 17271 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 ⊆ wss 3866 〈cop 4547 ↦ cmpt 5135 ran crn 5552 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 1st c1st 7759 2nd c2nd 7760 tpos ctpos 7967 Basecbs 16760 Hom chom 16813 Catccat 17167 Idccid 17168 Homf chomf 17169 oppCatcoppc 17214 Invcinv 17250 Isociso 17251 Func cfunc 17360 ∘func ccofu 17362 Nat cnat 17448 FuncCat cfuc 17449 SetCatcsetc 17581 ×c cxpc 17675 1stF c1stf 17676 2ndF c2ndf 17677 〈,〉F cprf 17678 evalF cevlf 17717 HomFchof 17756 Yoncyon 17757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-hom 16826 df-cco 16827 df-cat 17171 df-cid 17172 df-homf 17173 df-comf 17174 df-oppc 17215 df-sect 17252 df-inv 17253 df-iso 17254 df-ssc 17315 df-resc 17316 df-subc 17317 df-func 17364 df-cofu 17366 df-nat 17450 df-fuc 17451 df-setc 17582 df-xpc 17679 df-1stf 17680 df-2ndf 17681 df-prf 17682 df-evlf 17721 df-curf 17722 df-hof 17758 df-yon 17759 |
This theorem is referenced by: (None) |
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