Theorem yonffth 18508

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 2763	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2763	. 2 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		yonffth.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
5		yonffth.s	. 2 ⊢ 𝑆 = (SetCat‘𝑈)
6		eqid 2763	. 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))
7		yonffth.q	. 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		eqid 2763	. 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄)
9		eqid 2763	. 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))
10		eqid 2763	. 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11		eqid 2763	. 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 6929	. . . 4 ⊢ (Homf ‘𝑄) ∈ V
14	13	rnex 7956	. . 3 ⊢ ran (Homf ‘𝑄) ∈ V
15		yonffth.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
16		unexg 7792	. . 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 3936	. 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈))
20		eqid 2763	. 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 2763	. 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))))
22		eqid 2763	. 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 18506	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ⟨cop 4840   ↦ cmpt 5192  ran crn 5659  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  tpos ctpos 8280  Basecbs 17431  Hom chom 17483  Catccat 17888  Idccid 17889  Hom_f chomf 17890  oppCatcoppc 17935  Invcinv 17970   Func cfunc 18079   ∘_func ccofu 18081   Full cful 18129   Faith cfth 18130   Nat cnat 18169   FuncCat cfuc 18170  SetCatcsetc 18300   ×_c cxpc 18392   1^st_F c1stf 18393   2^nd_F c2ndf 18394   ⟨,⟩_F cprf 18395   eval_F cevlf 18433  Hom_Fchof 18472  Yoncyon 18473

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-hom 17497  df-cco 17498  df-cat 17892  df-cid 17893  df-homf 17894  df-comf 17895  df-oppc 17936  df-sect 17972  df-inv 17973  df-iso 17974  df-ssc 18035  df-resc 18036  df-subc 18037  df-func 18083  df-cofu 18085  df-full 18131  df-fth 18132  df-nat 18171  df-fuc 18172  df-setc 18301  df-xpc 18396  df-1stf 18397  df-2ndf 18398  df-prf 18399  df-evlf 18437  df-curf 18438  df-hof 18474  df-yon 18475

This theorem is used by:  yoniso  18509