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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 17230 | . 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅) |
3 | eqid 2821 | . 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 17522 | . . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
19 | 18 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
20 | funcrcl 17133 | . . . . 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 497 | . . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat) |
23 | 21 | simprd 498 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Cat) |
24 | 1, 22, 23 | fuccat 17240 | . 2 ⊢ (𝜑 → 𝑅 ∈ Cat) |
25 | 18 | simprd 498 | . 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 2821 | . . 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 17531 | . 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 17036 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 〈cop 4573 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 2nd c2nd 7688 tpos ctpos 7891 Basecbs 16483 Hom chom 16576 Catccat 16935 Idccid 16936 Homf chomf 16937 oppCatcoppc 16981 Invcinv 17015 Isociso 17016 Func cfunc 17124 ∘func ccofu 17126 Nat cnat 17211 FuncCat cfuc 17212 SetCatcsetc 17335 ×c cxpc 17418 1stF c1stf 17419 2ndF c2ndf 17420 〈,〉F cprf 17421 evalF cevlf 17459 HomFchof 17498 Yoncyon 17499 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-hom 16589 df-cco 16590 df-cat 16939 df-cid 16940 df-homf 16941 df-comf 16942 df-oppc 16982 df-sect 17017 df-inv 17018 df-iso 17019 df-ssc 17080 df-resc 17081 df-subc 17082 df-func 17128 df-cofu 17130 df-nat 17213 df-fuc 17214 df-setc 17336 df-xpc 17422 df-1stf 17423 df-2ndf 17424 df-prf 17425 df-evlf 17463 df-curf 17464 df-hof 17500 df-yon 17501 |
This theorem is referenced by: (None) |
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