Theorem yoneda 18282

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.)

Hypotheses

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)

Assertion

Ref	Expression
yoneda	⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥

Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝐼(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yoneda

Dummy variables 𝑔 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		yoneda.r	. . 3 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
2	1	fucbas 17958	. 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
3		eqid 2728	. 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 18271	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
19	18	simpld 493	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
20		funcrcl 17856	. . . . 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 493	. . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
23	21	simprd 494	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
24	1, 22, 23	fuccat 17969	. 2 ⊢ (𝜑 → 𝑅 ∈ Cat)
25	18	simprd 494	. 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 2728	. . 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 18280	. 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 17756	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∪ cun 3947   ⊆ wss 3949  ⟨cop 4638   ↦ cmpt 5235  ran crn 5683  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1^st c1st 7997  2^nd c2nd 7998  tpos ctpos 8237  Basecbs 17187  Hom chom 17251  Catccat 17651  Idccid 17652  Hom_f chomf 17653  oppCatcoppc 17698  Invcinv 17735  Isociso 17736   Func cfunc 17847   ∘_func ccofu 17849   Nat cnat 17938   FuncCat cfuc 17939  SetCatcsetc 18071   ×_c cxpc 18166   1^st_F c1stf 18167   2^nd_F c2ndf 18168   ⟨,⟩_F cprf 18169   eval_F cevlf 18208  Hom_Fchof 18247  Yoncyon 18248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-hom 17264  df-cco 17265  df-cat 17655  df-cid 17656  df-homf 17657  df-comf 17658  df-oppc 17699  df-sect 17737  df-inv 17738  df-iso 17739  df-ssc 17800  df-resc 17801  df-subc 17802  df-func 17851  df-cofu 17853  df-nat 17940  df-fuc 17941  df-setc 18072  df-xpc 18170  df-1stf 18171  df-2ndf 18172  df-prf 18173  df-evlf 18212  df-curf 18213  df-hof 18249  df-yon 18250

This theorem is referenced by: (None)