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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 2818 | . 2 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2818 | . 2 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | yonffth.o | . 2 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | yonffth.s | . 2 ⊢ 𝑆 = (SetCat‘𝑈) | |
6 | eqid 2818 | . 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) | |
7 | yonffth.q | . 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | eqid 2818 | . 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄) | |
9 | eqid 2818 | . 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) | |
10 | eqid 2818 | . 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆) | |
11 | eqid 2818 | . 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 6676 | . . . 4 ⊢ (Homf ‘𝑄) ∈ V | |
14 | 13 | rnex 7606 | . . 3 ⊢ ran (Homf ‘𝑄) ∈ V |
15 | yonffth.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
16 | unexg 7461 | . . 3 ⊢ ((ran (Homf ‘𝑄) ∈ V ∧ 𝑈 ∈ 𝑉) → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) | |
17 | 14, 15, 16 | sylancr 587 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) |
18 | yonffth.h | . 2 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
19 | ssidd 3987 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈)) | |
20 | eqid 2818 | . 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)))) | |
21 | eqid 2818 | . 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) | |
22 | eqid 2818 | . 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))))) | |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22 | yonffthlem 17520 | 1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 〈cop 4563 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7676 2nd c2nd 7677 tpos ctpos 7880 Basecbs 16471 Hom chom 16564 Catccat 16923 Idccid 16924 Homf chomf 16925 oppCatcoppc 16969 Invcinv 17003 Func cfunc 17112 ∘func ccofu 17114 Full cful 17160 Faith cfth 17161 Nat cnat 17199 FuncCat cfuc 17200 SetCatcsetc 17323 ×c cxpc 17406 1stF c1stf 17407 2ndF c2ndf 17408 〈,〉F cprf 17409 evalF cevlf 17447 HomFchof 17486 Yoncyon 17487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-homf 16929 df-comf 16930 df-oppc 16970 df-sect 17005 df-inv 17006 df-iso 17007 df-ssc 17068 df-resc 17069 df-subc 17070 df-func 17116 df-cofu 17118 df-full 17162 df-fth 17163 df-nat 17201 df-fuc 17202 df-setc 17324 df-xpc 17410 df-1stf 17411 df-2ndf 17412 df-prf 17413 df-evlf 17451 df-curf 17452 df-hof 17488 df-yon 17489 |
This theorem is referenced by: yoniso 17523 |
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