Theorem yonffth 18451

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 2761	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2761	. 2 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		yonffth.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
5		yonffth.s	. 2 ⊢ 𝑆 = (SetCat‘𝑈)
6		eqid 2761	. 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))
7		yonffth.q	. 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		eqid 2761	. 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄)
9		eqid 2761	. 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))
10		eqid 2761	. 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11		eqid 2761	. 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 6896	. . . 4 ⊢ (Homf ‘𝑄) ∈ V
14	13	rnex 7920	. . 3 ⊢ ran (Homf ‘𝑄) ∈ V
15		yonffth.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
16		unexg 7758	. . 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 3954	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈))
20		eqid 2761	. 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 2761	. 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))))
22		eqid 2761	. 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 18449	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590   ↦ cmpt 5186  ran crn 5652  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  2^nd c2nd 7998  tpos ctpos 8235  Basecbs 17380  Hom chom 17432  Catccat 17831  Idccid 17832  Hom_f chomf 17833  oppCatcoppc 17878  Invcinv 17913   Func cfunc 18022   ∘_func ccofu 18024   Full cful 18072   Faith cfth 18073   Nat cnat 18112   FuncCat cfuc 18113  SetCatcsetc 18243   ×_c cxpc 18335   1^st_F c1stf 18336   2^nd_F c2ndf 18337   ⟨,⟩_F cprf 18338   eval_F cevlf 18376  Hom_Fchof 18415  Yoncyon 18416

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-homf 17837  df-comf 17838  df-oppc 17879  df-sect 17915  df-inv 17916  df-iso 17917  df-ssc 17978  df-resc 17979  df-subc 17980  df-func 18026  df-cofu 18028  df-full 18074  df-fth 18075  df-nat 18114  df-fuc 18115  df-setc 18244  df-xpc 18339  df-1stf 18340  df-2ndf 18341  df-prf 18342  df-evlf 18380  df-curf 18381  df-hof 18417  df-yon 18418

This theorem is used by:  yoniso  18452