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Theorem yoneda 18232
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 ( βˆ’ , 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 π‘ˆ is used for forming the functor category 𝑄 = 𝐢 op β†’ SetCat(π‘ˆ), which itself does not (necessarily) live in π‘ˆ but instead is an element of the larger universe 𝑉. (If π‘ˆ is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set π‘ˆ = 𝑉 in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y π‘Œ = (Yonβ€˜πΆ)
yoneda.b 𝐡 = (Baseβ€˜πΆ)
yoneda.1 1 = (Idβ€˜πΆ)
yoneda.o 𝑂 = (oppCatβ€˜πΆ)
yoneda.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoneda.t 𝑇 = (SetCatβ€˜π‘‰)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomFβ€˜π‘„)
yoneda.r 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (πœ‘ β†’ 𝐢 ∈ Cat)
yoneda.w (πœ‘ β†’ 𝑉 ∈ π‘Š)
yoneda.u (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoneda.v (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
yoneda.i 𝐼 = (Isoβ€˜π‘…)
Assertion
Ref Expression
yoneda (πœ‘ β†’ 𝑀 ∈ (𝑍𝐼𝐸))
Distinct variable groups:   𝑓,π‘Ž,π‘₯, 1   𝐢,π‘Ž,𝑓,π‘₯   𝐸,π‘Ž,𝑓   𝐡,π‘Ž,𝑓,π‘₯   𝑂,π‘Ž,𝑓,π‘₯   𝑆,π‘Ž,𝑓,π‘₯   𝑄,π‘Ž,𝑓,π‘₯   𝑇,𝑓   πœ‘,π‘Ž,𝑓,π‘₯   π‘Œ,π‘Ž,𝑓,π‘₯   𝑍,π‘Ž,𝑓,π‘₯
Allowed substitution hints:   𝑅(π‘₯,𝑓,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑓,π‘Ž)   𝐸(π‘₯)   𝐻(π‘₯,𝑓,π‘Ž)   𝐼(π‘₯,𝑓,π‘Ž)   𝑀(π‘₯,𝑓,π‘Ž)   𝑉(π‘₯,𝑓,π‘Ž)   π‘Š(π‘₯,𝑓,π‘Ž)

Proof of Theorem yoneda
Dummy variables 𝑔 𝑦 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.r . . 3 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
21fucbas 17908 . 2 ((𝑄 Γ—c 𝑂) Func 𝑇) = (Baseβ€˜π‘…)
3 eqid 2732 . 2 (Invβ€˜π‘…) = (Invβ€˜π‘…)
4 yoneda.y . . . . . . 7 π‘Œ = (Yonβ€˜πΆ)
5 yoneda.b . . . . . . 7 𝐡 = (Baseβ€˜πΆ)
6 yoneda.1 . . . . . . 7 1 = (Idβ€˜πΆ)
7 yoneda.o . . . . . . 7 𝑂 = (oppCatβ€˜πΆ)
8 yoneda.s . . . . . . 7 𝑆 = (SetCatβ€˜π‘ˆ)
9 yoneda.t . . . . . . 7 𝑇 = (SetCatβ€˜π‘‰)
10 yoneda.q . . . . . . 7 𝑄 = (𝑂 FuncCat 𝑆)
11 yoneda.h . . . . . . 7 𝐻 = (HomFβ€˜π‘„)
12 yoneda.e . . . . . . 7 𝐸 = (𝑂 evalF 𝑆)
13 yoneda.z . . . . . . 7 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
14 yoneda.c . . . . . . 7 (πœ‘ β†’ 𝐢 ∈ Cat)
15 yoneda.w . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ π‘Š)
16 yoneda.u . . . . . . 7 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
17 yoneda.v . . . . . . 7 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
184, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17yonedalem1 18221 . . . . . 6 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
1918simpld 495 . . . . 5 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
20 funcrcl 17809 . . . . 5 (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) β†’ ((𝑄 Γ—c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
2119, 20syl 17 . . . 4 (πœ‘ β†’ ((𝑄 Γ—c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
2221simpld 495 . . 3 (πœ‘ β†’ (𝑄 Γ—c 𝑂) ∈ Cat)
2321simprd 496 . . 3 (πœ‘ β†’ 𝑇 ∈ Cat)
241, 22, 23fuccat 17919 . 2 (πœ‘ β†’ 𝑅 ∈ Cat)
2518simprd 496 . 2 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
26 yoneda.i . 2 𝐼 = (Isoβ€˜π‘…)
27 yoneda.m . . 3 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
28 eqid 2732 . . 3 (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’))))) = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))))
294, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 27, 3, 28yonedainv 18230 . 2 (πœ‘ β†’ 𝑀(𝑍(Invβ€˜π‘…)𝐸)(𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’))))))
302, 3, 24, 19, 25, 26, 29inviso1 17709 1 (πœ‘ β†’ 𝑀 ∈ (𝑍𝐼𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3945   βŠ† wss 3947  βŸ¨cop 4633   ↦ cmpt 5230  ran crn 5676  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7969  2nd c2nd 7970  tpos ctpos 8206  Basecbs 17140  Hom chom 17204  Catccat 17604  Idccid 17605  Homf chomf 17606  oppCatcoppc 17651  Invcinv 17688  Isociso 17689   Func cfunc 17800   ∘func ccofu 17802   Nat cnat 17888   FuncCat cfuc 17889  SetCatcsetc 18021   Γ—c cxpc 18116   1stF c1stf 18117   2ndF c2ndf 18118   ⟨,⟩F cprf 18119   evalF cevlf 18158  HomFchof 18197  Yoncyon 18198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-hom 17217  df-cco 17218  df-cat 17608  df-cid 17609  df-homf 17610  df-comf 17611  df-oppc 17652  df-sect 17690  df-inv 17691  df-iso 17692  df-ssc 17753  df-resc 17754  df-subc 17755  df-func 17804  df-cofu 17806  df-nat 17890  df-fuc 17891  df-setc 18022  df-xpc 18120  df-1stf 18121  df-2ndf 18122  df-prf 18123  df-evlf 18162  df-curf 18163  df-hof 18199  df-yon 18200
This theorem is referenced by: (None)
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