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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 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 (Homf ‘𝑄) ∪ 𝑈)) = (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)) | |
7 | yonffth.q | . 2 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | eqid 2732 | . 2 ⊢ (HomF‘𝑄) = (HomF‘𝑄) | |
9 | eqid 2732 | . 2 ⊢ ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) = ((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈))) | |
10 | eqid 2732 | . 2 ⊢ (𝑂 evalF 𝑆) = (𝑂 evalF 𝑆) | |
11 | eqid 2732 | . 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 6892 | . . . 4 ⊢ (Homf ‘𝑄) ∈ V | |
14 | 13 | rnex 7887 | . . 3 ⊢ ran (Homf ‘𝑄) ∈ V |
15 | yonffth.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
16 | unexg 7720 | . . 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 4002 | . 2 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ (ran (Homf ‘𝑄) ∪ 𝑈)) | |
20 | eqid 2732 | . 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 2732 | . 2 ⊢ (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) = (Inv‘((𝑄 ×c 𝑂) FuncCat (SetCat‘(ran (Homf ‘𝑄) ∪ 𝑈)))) | |
22 | eqid 2732 | . 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 18219 | 1 ⊢ (𝜑 → 𝑌 ∈ ((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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