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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 2820 | . 2 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2820 | . 2 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
4 | yonffth.o | . 2 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | yonffth.s | . 2 ⊢ 𝑆 = (SetCat‘𝑈) | |
6 | eqid 2820 | . 2 ⊢ (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) | |
7 | yonffth.q | . 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | eqid 2820 | . 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄) | |
9 | eqid 2820 | . 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) | |
10 | eqid 2820 | . 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆) | |
11 | eqid 2820 | . 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 7610 | . . 3 ⊢ ran (Homf ‘𝑄) ∈ V |
15 | yonffth.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
16 | unexg 7465 | . . 3 ⊢ ((ran (Homf ‘𝑄) ∈ V ∧ 𝑈 ∈ 𝑉) → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) | |
17 | 14, 15, 16 | sylancr 589 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ∈ V) |
18 | yonffth.h | . 2 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
19 | ssidd 3983 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈)) | |
20 | eqid 2820 | . 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 2820 | . 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) | |
22 | eqid 2820 | . 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 17527 | 1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 ∩ cin 3928 ⊆ wss 3929 〈cop 4566 ↦ cmpt 5139 ran crn 5549 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 1st c1st 7680 2nd c2nd 7681 tpos ctpos 7884 Basecbs 16478 Hom chom 16571 Catccat 16930 Idccid 16931 Homf chomf 16932 oppCatcoppc 16976 Invcinv 17010 Func cfunc 17119 ∘func ccofu 17121 Full cful 17167 Faith cfth 17168 Nat cnat 17206 FuncCat cfuc 17207 SetCatcsetc 17330 ×c cxpc 17413 1stF c1stf 17414 2ndF c2ndf 17415 〈,〉F cprf 17416 evalF cevlf 17454 HomFchof 17493 Yoncyon 17494 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-hom 16584 df-cco 16585 df-cat 16934 df-cid 16935 df-homf 16936 df-comf 16937 df-oppc 16977 df-sect 17012 df-inv 17013 df-iso 17014 df-ssc 17075 df-resc 17076 df-subc 17077 df-func 17123 df-cofu 17125 df-full 17169 df-fth 17170 df-nat 17208 df-fuc 17209 df-setc 17331 df-xpc 17417 df-1stf 17418 df-2ndf 17419 df-prf 17420 df-evlf 17458 df-curf 17459 df-hof 17495 df-yon 17496 |
This theorem is referenced by: yoniso 17530 |
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