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Theorem yoneda 18324
Description: The Yoneda Lemma. There is a natural isomorphism between the functors 𝑍 and 𝐸, where 𝑍(𝐹, 𝑋) is the natural transformations from Yon(𝑋) = Hom ( − , 𝑋) to 𝐹, and 𝐸(𝐹, 𝑋) = 𝐹(𝑋) is the evaluation functor. Here we need two universes to state the claim: the smaller universe 𝑈 is used for forming the functor category 𝑄 = 𝐶 op → SetCat(𝑈), which itself does not (necessarily) live in 𝑈 but instead is an element of the larger universe 𝑉. (If 𝑈 is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set 𝑈 = 𝑉 in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
yoneda.i 𝐼 = (Iso‘𝑅)
Assertion
Ref Expression
yoneda (𝜑𝑀 ∈ (𝑍𝐼𝐸))
Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝐼(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yoneda
Dummy variables 𝑔 𝑦 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.r . . 3 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
21fucbas 18004 . 2 ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
3 eqid 2736 . 2 (Inv‘𝑅) = (Inv‘𝑅)
4 yoneda.y . . . . . . 7 𝑌 = (Yon‘𝐶)
5 yoneda.b . . . . . . 7 𝐵 = (Base‘𝐶)
6 yoneda.1 . . . . . . 7 1 = (Id‘𝐶)
7 yoneda.o . . . . . . 7 𝑂 = (oppCat‘𝐶)
8 yoneda.s . . . . . . 7 𝑆 = (SetCat‘𝑈)
9 yoneda.t . . . . . . 7 𝑇 = (SetCat‘𝑉)
10 yoneda.q . . . . . . 7 𝑄 = (𝑂 FuncCat 𝑆)
11 yoneda.h . . . . . . 7 𝐻 = (HomF𝑄)
12 yoneda.e . . . . . . 7 𝐸 = (𝑂 evalF 𝑆)
13 yoneda.z . . . . . . 7 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
14 yoneda.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
15 yoneda.w . . . . . . 7 (𝜑𝑉𝑊)
16 yoneda.u . . . . . . 7 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
17 yoneda.v . . . . . . 7 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
184, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17yonedalem1 18313 . . . . . 6 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
1918simpld 494 . . . . 5 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
20 funcrcl 17904 . . . . 5 (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
2119, 20syl 17 . . . 4 (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
2221simpld 494 . . 3 (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
2321simprd 495 . . 3 (𝜑𝑇 ∈ Cat)
241, 22, 23fuccat 18014 . 2 (𝜑𝑅 ∈ Cat)
2518simprd 495 . 2 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
26 yoneda.i . 2 𝐼 = (Iso‘𝑅)
27 yoneda.m . . 3 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
28 eqid 2736 . . 3 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
294, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 27, 3, 28yonedainv 18322 . 2 (𝜑𝑀(𝑍(Inv‘𝑅)𝐸)(𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢))))))
302, 3, 24, 19, 25, 26, 29inviso1 17806 1 (𝜑𝑀 ∈ (𝑍𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3948  wss 3950  cop 4630  cmpt 5223  ran crn 5684  cfv 6559  (class class class)co 7429  cmpo 7431  1st c1st 8008  2nd c2nd 8009  tpos ctpos 8246  Basecbs 17243  Hom chom 17304  Catccat 17703  Idccid 17704  Homf chomf 17705  oppCatcoppc 17750  Invcinv 17785  Isociso 17786   Func cfunc 17895  func ccofu 17897   Nat cnat 17985   FuncCat cfuc 17986  SetCatcsetc 18116   ×c cxpc 18209   1stF c1stf 18210   2ndF c2ndf 18211   ⟨,⟩F cprf 18212   evalF cevlf 18250  HomFchof 18289  Yoncyon 18290
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-tpos 8247  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-er 8741  df-map 8864  df-pm 8865  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-nn 12263  df-2 12325  df-3 12326  df-4 12327  df-5 12328  df-6 12329  df-7 12330  df-8 12331  df-9 12332  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12875  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17244  df-ress 17271  df-hom 17317  df-cco 17318  df-cat 17707  df-cid 17708  df-homf 17709  df-comf 17710  df-oppc 17751  df-sect 17787  df-inv 17788  df-iso 17789  df-ssc 17850  df-resc 17851  df-subc 17852  df-func 17899  df-cofu 17901  df-nat 17987  df-fuc 17988  df-setc 18117  df-xpc 18213  df-1stf 18214  df-2ndf 18215  df-prf 18216  df-evlf 18254  df-curf 18255  df-hof 18291  df-yon 18292
This theorem is referenced by: (None)
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