Theorem yoneda 18457

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 18138	. 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
3		eqid 2760	. 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 18446	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
19	18	simpld 500	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
20		funcrcl 18038	. . . . 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 500	. . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
23	21	simprd 501	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
24	1, 22, 23	fuccat 18148	. 2 ⊢ (𝜑 → 𝑅 ∈ Cat)
25	18	simprd 501	. 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 2760	. . 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 18455	. 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 17941	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ⟨cop 4589   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  tpos ctpos 8242  Basecbs 17387  Hom chom 17439  Catccat 17838  Idccid 17839  Hom_f chomf 17840  oppCatcoppc 17885  Invcinv 17920  Isociso 17921   Func cfunc 18029   ∘_func ccofu 18031   Nat cnat 18119   FuncCat cfuc 18120  SetCatcsetc 18250   ×_c cxpc 18342   1^st_F c1stf 18343   2^nd_F c2ndf 18344   ⟨,⟩_F cprf 18345   eval_F cevlf 18383  Hom_Fchof 18422  Yoncyon 18423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-homf 17844  df-comf 17845  df-oppc 17886  df-sect 17922  df-inv 17923  df-iso 17924  df-ssc 17985  df-resc 17986  df-subc 17987  df-func 18033  df-cofu 18035  df-nat 18121  df-fuc 18122  df-setc 18251  df-xpc 18346  df-1stf 18347  df-2ndf 18348  df-prf 18349  df-evlf 18387  df-curf 18388  df-hof 18424  df-yon 18425

This theorem is used by: (None)