Theorem yonffth 18245

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 2729	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2729	. 2 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		yonffth.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
5		yonffth.s	. 2 ⊢ 𝑆 = (SetCat‘𝑈)
6		eqid 2729	. 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))
7		yonffth.q	. 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		eqid 2729	. 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄)
9		eqid 2729	. 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))
10		eqid 2729	. 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11		eqid 2729	. 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 6871	. . . 4 ⊢ (Homf ‘𝑄) ∈ V
14	13	rnex 7886	. . 3 ⊢ ran (Homf ‘𝑄) ∈ V
15		yonffth.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
16		unexg 7719	. . 3 ⊢ ((ran (Homf ‘𝑄) ∈ V ∧ 𝑈 ∈ 𝑉) → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V)
17	14, 15, 16	sylancr 587	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V)
18		yonffth.h	. 2 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
19		ssidd 3970	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈))
20		eqid 2729	. 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 2729	. 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))))
22		eqid 2729	. 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 18243	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ⟨cop 4595   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  tpos ctpos 8204  Basecbs 17179  Hom chom 17231  Catccat 17625  Idccid 17626  Hom_f chomf 17627  oppCatcoppc 17672  Invcinv 17707   Func cfunc 17816   ∘_func ccofu 17818   Full cful 17866   Faith cfth 17867   Nat cnat 17906   FuncCat cfuc 17907  SetCatcsetc 18037   ×_c cxpc 18129   1^st_F c1stf 18130   2^nd_F c2ndf 18131   ⟨,⟩_F cprf 18132   eval_F cevlf 18170  Hom_Fchof 18209  Yoncyon 18210

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-homf 17631  df-comf 17632  df-oppc 17673  df-sect 17709  df-inv 17710  df-iso 17711  df-ssc 17772  df-resc 17773  df-subc 17774  df-func 17820  df-cofu 17822  df-full 17868  df-fth 17869  df-nat 17908  df-fuc 17909  df-setc 18038  df-xpc 18133  df-1stf 18134  df-2ndf 18135  df-prf 18136  df-evlf 18174  df-curf 18175  df-hof 18211  df-yon 18212

This theorem is referenced by:  yoniso  18246