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Mirrors > Home > MPE Home > Th. List > yonffth | Structured version Visualization version GIF version |
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.) |
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 | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yonffth.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
yonffth.h | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
Ref | Expression |
---|---|
yonffth | ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yonffth.y | . 2 ⊢ 𝑌 = (Yon‘𝐶) | |
2 | eqid 2610 | . 2 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2610 | . 2 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | yonffth.o | . 2 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | yonffth.s | . 2 ⊢ 𝑆 = (SetCat‘𝑈) | |
6 | eqid 2610 | . 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) | |
7 | yonffth.q | . 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | eqid 2610 | . 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄) | |
9 | eqid 2610 | . 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) | |
10 | eqid 2610 | . 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆) | |
11 | eqid 2610 | . 2 ⊢ ((HomF‘𝑄) ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) = ((HomF‘𝑄) ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) | |
12 | yonffth.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
13 | fvex 6113 | . . . 4 ⊢ (Homf ‘𝑄) ∈ V | |
14 | 13 | rnex 6992 | . . 3 ⊢ ran (Homf ‘𝑄) ∈ V |
15 | yonffth.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
16 | unexg 6857 | . . 3 ⊢ ((ran (Homf ‘𝑄) ∈ V ∧ 𝑈 ∈ 𝑉) → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) | |
17 | 14, 15, 16 | sylancr 694 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) |
18 | yonffth.h | . 2 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
19 | ssid 3587 | . . 3 ⊢ (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈) | |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈)) |
21 | eqid 2610 | . 2 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘((Id‘𝐶)‘𝑥)))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘((Id‘𝐶)‘𝑥)))) | |
22 | eqid 2610 | . 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) | |
23 | eqid 2610 | . 2 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ (Base‘𝐶) ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))) | |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 20, 21, 22, 23 | yonffthlem 16745 | 1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 tpos ctpos 7238 Basecbs 15695 Hom chom 15779 Catccat 16148 Idccid 16149 Homf chomf 16150 oppCatcoppc 16194 Invcinv 16228 Func cfunc 16337 ∘func ccofu 16339 Full cful 16385 Faith cfth 16386 Nat cnat 16424 FuncCat cfuc 16425 SetCatcsetc 16548 ×c cxpc 16631 1stF c1stf 16632 2ndF c2ndf 16633 〈,〉F cprf 16634 evalF cevlf 16672 HomFchof 16711 Yoncyon 16712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-hom 15793 df-cco 15794 df-cat 16152 df-cid 16153 df-homf 16154 df-comf 16155 df-oppc 16195 df-sect 16230 df-inv 16231 df-iso 16232 df-ssc 16293 df-resc 16294 df-subc 16295 df-func 16341 df-cofu 16343 df-full 16387 df-fth 16388 df-nat 16426 df-fuc 16427 df-setc 16549 df-xpc 16635 df-1stf 16636 df-2ndf 16637 df-prf 16638 df-evlf 16676 df-curf 16677 df-hof 16713 df-yon 16714 |
This theorem is referenced by: yoniso 16748 |
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