Theorem yonffth 18208

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 6839	. . . 4 ⊢ (Homf ‘𝑄) ∈ V
14	13	rnex 7850	. . 3 ⊢ ran (Homf ‘𝑄) ∈ V
15		yonffth.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
16		unexg 7683	. . 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 3961	. 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 18206	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4585   ↦ cmpt 5176  ran crn 5624  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  tpos ctpos 8165  Basecbs 17138  Hom chom 17190  Catccat 17588  Idccid 17589  Hom_f chomf 17590  oppCatcoppc 17635  Invcinv 17670   Func cfunc 17779   ∘_func ccofu 17781   Full cful 17829   Faith cfth 17830   Nat cnat 17869   FuncCat cfuc 17870  SetCatcsetc 18000   ×_c cxpc 18092   1^st_F c1stf 18093   2^nd_F c2ndf 18094   ⟨,⟩_F cprf 18095   eval_F cevlf 18133  Hom_Fchof 18172  Yoncyon 18173

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-pm 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-hom 17203  df-cco 17204  df-cat 17592  df-cid 17593  df-homf 17594  df-comf 17595  df-oppc 17636  df-sect 17672  df-inv 17673  df-iso 17674  df-ssc 17735  df-resc 17736  df-subc 17737  df-func 17783  df-cofu 17785  df-full 17831  df-fth 17832  df-nat 17871  df-fuc 17872  df-setc 18001  df-xpc 18096  df-1stf 18097  df-2ndf 18098  df-prf 18099  df-evlf 18137  df-curf 18138  df-hof 18174  df-yon 18175

This theorem is referenced by:  yoniso  18209