Theorem yoneda 18418

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 18099	. 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 18407	. . . . . 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 17999	. . . . 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 18109	. 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 18416	. 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 17902	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3896   ⊆ wss 3898  ⟨cop 4589   ↦ cmpt 5185  ran crn 5648  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7982  2^nd c2nd 7983  tpos ctpos 8220  Basecbs 17348  Hom chom 17400  Catccat 17799  Idccid 17800  Hom_f chomf 17801  oppCatcoppc 17846  Invcinv 17881  Isociso 17882   Func cfunc 17990   ∘_func ccofu 17992   Nat cnat 18080   FuncCat cfuc 18081  SetCatcsetc 18211   ×_c cxpc 18303   1^st_F c1stf 18304   2^nd_F c2ndf 18305   ⟨,⟩_F cprf 18306   eval_F cevlf 18344  Hom_Fchof 18383  Yoncyon 18384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-hom 17413  df-cco 17414  df-cat 17803  df-cid 17804  df-homf 17805  df-comf 17806  df-oppc 17847  df-sect 17883  df-inv 17884  df-iso 17885  df-ssc 17946  df-resc 17947  df-subc 17948  df-func 17994  df-cofu 17996  df-nat 18082  df-fuc 18083  df-setc 18212  df-xpc 18307  df-1stf 18308  df-2ndf 18309  df-prf 18310  df-evlf 18348  df-curf 18349  df-hof 18385  df-yon 18386

This theorem is used by: (None)