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Theorem yonffth 18221
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 2732 . 2 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2732 . 2 (Id‘𝐶) = (Id‘𝐶)
4 yonffth.o . 2 𝑂 = (oppCat‘𝐶)
5 yonffth.s . 2 𝑆 = (SetCat‘𝑈)
6 eqid 2732 . 2 (SetCat‘(ran (Homf𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf𝑄) ∪ 𝑈))
7 yonffth.q . 2 𝑄 = (𝑂 FuncCat 𝑆)
8 eqid 2732 . 2 (HomF𝑄) = (HomF𝑄)
9 eqid 2732 . 2 ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))
10 eqid 2732 . 2 (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆)
11 eqid 2732 . 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 6892 . . . 4 (Homf𝑄) ∈ V
1413rnex 7887 . . 3 ran (Homf𝑄) ∈ V
15 yonffth.u . . 3 (𝜑𝑈𝑉)
16 unexg 7720 . . 3 ((ran (Homf𝑄) ∈ V ∧ 𝑈𝑉) → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
1714, 15, 16sylancr 587 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ∈ V)
18 yonffth.h . 2 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
19 ssidd 4002 . 2 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ (ran (Homf𝑄) ∪ 𝑈))
20 eqid 2732 . 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))))
21 eqid 2732 . 2 (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf𝑄) ∪ 𝑈))))
22 eqid 2732 . 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)))))
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22yonffthlem 18219 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  cun 3943  cin 3944  wss 3945  cop 4629  cmpt 5225  ran crn 5671  cfv 6533  (class class class)co 7394  cmpo 7396  1st c1st 7957  2nd c2nd 7958  tpos ctpos 8194  Basecbs 17128  Hom chom 17192  Catccat 17592  Idccid 17593  Homf chomf 17594  oppCatcoppc 17639  Invcinv 17676   Func cfunc 17788  func ccofu 17790   Full cful 17837   Faith cfth 17838   Nat cnat 17876   FuncCat cfuc 17877  SetCatcsetc 18009   ×c cxpc 18104   1stF c1stf 18105   2ndF c2ndf 18106   ⟨,⟩F cprf 18107   evalF cevlf 18146  HomFchof 18185  Yoncyon 18186
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171
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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-1st 7959  df-2nd 7960  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-er 8688  df-map 8807  df-pm 8808  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-hom 17205  df-cco 17206  df-cat 17596  df-cid 17597  df-homf 17598  df-comf 17599  df-oppc 17640  df-sect 17678  df-inv 17679  df-iso 17680  df-ssc 17741  df-resc 17742  df-subc 17743  df-func 17792  df-cofu 17794  df-full 17839  df-fth 17840  df-nat 17878  df-fuc 17879  df-setc 18010  df-xpc 18108  df-1stf 18109  df-2ndf 18110  df-prf 18111  df-evlf 18150  df-curf 18151  df-hof 18187  df-yon 18188
This theorem is referenced by:  yoniso  18222
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