Theorem yoneda 18250

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 17931	. 2 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
3		eqid 2730	. 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 18239	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
19	18	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
20		funcrcl 17831	. . . . 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 494	. . 3 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
23	21	simprd 495	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
24	1, 22, 23	fuccat 17941	. 2 ⊢ (𝜑 → 𝑅 ∈ Cat)
25	18	simprd 495	. 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 2730	. . 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 18248	. 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 17734	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐼𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3914   ⊆ wss 3916  ⟨cop 4597   ↦ cmpt 5190  ran crn 5641  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206  Basecbs 17185  Hom chom 17237  Catccat 17631  Idccid 17632  Hom_f chomf 17633  oppCatcoppc 17678  Invcinv 17713  Isociso 17714   Func cfunc 17822   ∘_func ccofu 17824   Nat cnat 17912   FuncCat cfuc 17913  SetCatcsetc 18043   ×_c cxpc 18135   1^st_F c1stf 18136   2^nd_F c2ndf 18137   ⟨,⟩_F cprf 18138   eval_F cevlf 18176  Hom_Fchof 18215  Yoncyon 18216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-pm 8804  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-ress 17207  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-homf 17637  df-comf 17638  df-oppc 17679  df-sect 17715  df-inv 17716  df-iso 17717  df-ssc 17778  df-resc 17779  df-subc 17780  df-func 17826  df-cofu 17828  df-nat 17914  df-fuc 17915  df-setc 18044  df-xpc 18139  df-1stf 18140  df-2ndf 18141  df-prf 18142  df-evlf 18180  df-curf 18181  df-hof 18217  df-yon 18218

This theorem is referenced by: (None)