Theorem yoneda 18345

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 18026	. 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
3		eqid 2762	. 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 18334	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
19	18	simpld 499	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
20		funcrcl 17926	. . . . 5 ⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
21	19, 20	syl 18	. . . 4 ⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
22	21	simpld 499	. . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
23	21	simprd 500	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
24	1, 22, 23	fuccat 18036	. 2 ⊢ (𝜑 → 𝑅 ∈ Cat)
25	18	simprd 500	. 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 2762	. . 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 18343	. 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 17829	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ∪ cun 3902   ⊆ wss 3904  ⟨cop 4594   ↦ cmpt 5191  ran crn 5661  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7982  2^nd c2nd 7983  tpos ctpos 8219  Basecbs 17275  Hom chom 17327  Catccat 17726  Idccid 17727  Hom_f chomf 17728  oppCatcoppc 17773  Invcinv 17808  Isociso 17809   Func cfunc 17917   ∘_func ccofu 17919   Nat cnat 18007   FuncCat cfuc 18008  SetCatcsetc 18138   ×_c cxpc 18230   1^st_F c1stf 18231   2^nd_F c2ndf 18232   ⟨,⟩_F cprf 18233   eval_F cevlf 18271  Hom_Fchof 18310  Yoncyon 18311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8220  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-hom 17340  df-cco 17341  df-cat 17730  df-cid 17731  df-homf 17732  df-comf 17733  df-oppc 17774  df-sect 17810  df-inv 17811  df-iso 17812  df-ssc 17873  df-resc 17874  df-subc 17875  df-func 17921  df-cofu 17923  df-nat 18009  df-fuc 18010  df-setc 18139  df-xpc 18234  df-1stf 18235  df-2ndf 18236  df-prf 18237  df-evlf 18275  df-curf 18276  df-hof 18312  df-yon 18313

This theorem is used by: (None)