MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonffth Structured version   Visualization version   GIF version

Theorem yonffth 17125
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 (𝜑𝐶 ∈ Cat)
yonffth.u (𝜑𝑈𝑉)
yonffth.h (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
Assertion
Ref Expression
yonffth (𝜑𝑌 ∈ ((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.
StepHypRef Expression
1 yonffth.y . 2 𝑌 = (Yon‘𝐶)
2 eqid 2760 . 2 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2760 . 2 (Id‘𝐶) = (Id‘𝐶)
4 yonffth.o . 2 𝑂 = (oppCat‘𝐶)
5 yonffth.s . 2 𝑆 = (SetCat‘𝑈)
6 eqid 2760 . 2 (SetCat‘(ran (Homf𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf𝑄) ∪ 𝑈))
7 yonffth.q . 2 𝑄 = (𝑂 FuncCat 𝑆)
8 eqid 2760 . 2 (HomF𝑄) = (HomF𝑄)
9 eqid 2760 . 2 ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))
10 eqid 2760 . 2 (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11 eqid 2760 . 2 ((HomF𝑄) ∘func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))) = ((HomF𝑄) ∘func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12 yonffth.c . 2 (𝜑𝐶 ∈ Cat)
13 fvex 6362 . . . 4 (Homf𝑄) ∈ V
1413rnex 7265 . . 3 ran (Homf𝑄) ∈ V
15 yonffth.u . . 3 (𝜑𝑈𝑉)
16 unexg 7124 . . 3 ((ran (Homf𝑄) ∈ V ∧ 𝑈𝑉) → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
1714, 15, 16sylancr 698 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
18 yonffth.h . 2 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
19 ssid 3765 . . 3 (ran (Homf𝑄) ∪ 𝑈) ⊆ (ran (Homf𝑄) ∪ 𝑈)
2019a1i 11 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ (ran (Homf𝑄) ∪ 𝑈))
21 eqid 2760 . 2 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘((Id‘𝐶)‘𝑥)))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘((Id‘𝐶)‘𝑥))))
22 eqid 2760 . 2 (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))))
23 eqid 2760 . 2 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 20, 21, 22, 23yonffthlem 17123 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  cin 3714  wss 3715  cop 4327  cmpt 4881  ran crn 5267  cfv 6049  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  tpos ctpos 7520  Basecbs 16059  Hom chom 16154  Catccat 16526  Idccid 16527  Homf chomf 16528  oppCatcoppc 16572  Invcinv 16606   Func cfunc 16715  func ccofu 16717   Full cful 16763   Faith cfth 16764   Nat cnat 16802   FuncCat cfuc 16803  SetCatcsetc 16926   ×c cxpc 17009   1stF c1stf 17010   2ndF c2ndf 17011   ⟨,⟩F cprf 17012   evalF cevlf 17050  HomFchof 17089  Yoncyon 17090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-hom 16168  df-cco 16169  df-cat 16530  df-cid 16531  df-homf 16532  df-comf 16533  df-oppc 16573  df-sect 16608  df-inv 16609  df-iso 16610  df-ssc 16671  df-resc 16672  df-subc 16673  df-func 16719  df-cofu 16721  df-full 16765  df-fth 16766  df-nat 16804  df-fuc 16805  df-setc 16927  df-xpc 17013  df-1stf 17014  df-2ndf 17015  df-prf 17016  df-evlf 17054  df-curf 17055  df-hof 17091  df-yon 17092
This theorem is referenced by:  yoniso  17126
  Copyright terms: Public domain W3C validator