Theorem yonffth 18362

Description: The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category 𝐶 as a full subcategory of the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 29-Jan-2017.)

Hypotheses

Ref	Expression
yonffth.y	⊢ 𝑌 = (Yon‘𝐶)
yonffth.o	⊢ 𝑂 = (oppCat‘𝐶)
yonffth.s	⊢ 𝑆 = (SetCat‘𝑈)
yonffth.q	⊢ 𝑄 = (𝑂 FuncCat 𝑆)
yonffth.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yonffth.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
yonffth.h	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)

Assertion

Ref	Expression
yonffth	⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Proof of Theorem yonffth

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

Step	Hyp	Ref	Expression
1		yonffth.y	. 2 ⊢ 𝑌 = (Yon‘𝐶)
2		eqid 2765	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2765	. 2 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		yonffth.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
5		yonffth.s	. 2 ⊢ 𝑆 = (SetCat‘𝑈)
6		eqid 2765	. 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))
7		yonffth.q	. 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		eqid 2765	. 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄)
9		eqid 2765	. 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))
10		eqid 2765	. 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11		eqid 2765	. 2 ⊢ ((HomF‘𝑄) ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))) = ((HomF‘𝑄) ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12		yonffth.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		fvex 6898	. . . 4 ⊢ (Homf ‘𝑄) ∈ V
14	13	rnex 7913	. . 3 ⊢ ran (Homf ‘𝑄) ∈ V
15		yonffth.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
16		unexg 7751	. . 3 ⊢ ((ran (Homf ‘𝑄) ∈ V ∧ 𝑈 ∈ 𝑉) → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V)
17	14, 15, 16	sylancr 599	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V)
18		yonffth.h	. 2 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
19		ssidd 3961	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈))
20		eqid 2765	. 2 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘((Id‘𝐶)‘𝑥)))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘((Id‘𝐶)‘𝑥))))
21		eqid 2765	. 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))))
22		eqid 2765	. 2 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22	yonffthlem 18360	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ⟨cop 4597   ↦ cmpt 5194  ran crn 5664  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  tpos ctpos 8227  Basecbs 17291  Hom chom 17343  Catccat 17742  Idccid 17743  Hom_f chomf 17744  oppCatcoppc 17789  Invcinv 17824   Func cfunc 17933   ∘_func ccofu 17935   Full cful 17983   Faith cfth 17984   Nat cnat 18023   FuncCat cfuc 18024  SetCatcsetc 18154   ×_c cxpc 18246   1^st_F c1stf 18247   2^nd_F c2ndf 18248   ⟨,⟩_F cprf 18249   eval_F cevlf 18287  Hom_Fchof 18326  Yoncyon 18327

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-hom 17356  df-cco 17357  df-cat 17746  df-cid 17747  df-homf 17748  df-comf 17749  df-oppc 17790  df-sect 17826  df-inv 17827  df-iso 17828  df-ssc 17889  df-resc 17890  df-subc 17891  df-func 17937  df-cofu 17939  df-full 17985  df-fth 17986  df-nat 18025  df-fuc 18026  df-setc 18155  df-xpc 18250  df-1stf 18251  df-2ndf 18252  df-prf 18253  df-evlf 18291  df-curf 18292  df-hof 18328  df-yon 18329

This theorem is used by:  yoniso  18363