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Theorem yonffth 17131
 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 2770 . 2 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2770 . 2 (Id‘𝐶) = (Id‘𝐶)
4 yonffth.o . 2 𝑂 = (oppCat‘𝐶)
5 yonffth.s . 2 𝑆 = (SetCat‘𝑈)
6 eqid 2770 . 2 (SetCat‘(ran (Homf𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf𝑄) ∪ 𝑈))
7 yonffth.q . 2 𝑄 = (𝑂 FuncCat 𝑆)
8 eqid 2770 . 2 (HomF𝑄) = (HomF𝑄)
9 eqid 2770 . 2 ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))
10 eqid 2770 . 2 (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11 eqid 2770 . 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 6342 . . . 4 (Homf𝑄) ∈ V
1413rnex 7246 . . 3 ran (Homf𝑄) ∈ V
15 yonffth.u . . 3 (𝜑𝑈𝑉)
16 unexg 7105 . . 3 ((ran (Homf𝑄) ∈ V ∧ 𝑈𝑉) → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
1714, 15, 16sylancr 567 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
18 yonffth.h . 2 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
19 ssid 3771 . . 3 (ran (Homf𝑄) ∪ 𝑈) ⊆ (ran (Homf𝑄) ∪ 𝑈)
2019a1i 11 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ (ran (Homf𝑄) ∪ 𝑈))
21 eqid 2770 . 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 2770 . 2 (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))))
23 eqid 2770 . 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 17129 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∪ cun 3719   ∩ cin 3720   ⊆ wss 3721  ⟨cop 4320   ↦ cmpt 4861  ran crn 5250  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  1st c1st 7312  2nd c2nd 7313  tpos ctpos 7502  Basecbs 16063  Hom chom 16159  Catccat 16531  Idccid 16532  Homf chomf 16533  oppCatcoppc 16577  Invcinv 16611   Func cfunc 16720   ∘func ccofu 16722   Full cful 16768   Faith cfth 16769   Nat cnat 16807   FuncCat cfuc 16808  SetCatcsetc 16931   ×c cxpc 17015   1stF c1stf 17016   2ndF c2ndf 17017   ⟨,⟩F cprf 17018   evalF cevlf 17056  HomFchof 17095  Yoncyon 17096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-hom 16173  df-cco 16174  df-cat 16535  df-cid 16536  df-homf 16537  df-comf 16538  df-oppc 16578  df-sect 16613  df-inv 16614  df-iso 16615  df-ssc 16676  df-resc 16677  df-subc 16678  df-func 16724  df-cofu 16726  df-full 16770  df-fth 16771  df-nat 16809  df-fuc 16810  df-setc 16932  df-xpc 17019  df-1stf 17020  df-2ndf 17021  df-prf 17022  df-evlf 17060  df-curf 17061  df-hof 17097  df-yon 17098 This theorem is referenced by:  yoniso  17132
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