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Mirrors > Home > MPE Home > Th. List > yoncl | Structured version Visualization version GIF version |
Description: The Yoneda embedding is a functor from the category to the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 17-Jan-2017.) |
Ref | Expression |
---|---|
yonval.y | ⊢ 𝑌 = (Yon‘𝐶) |
yonval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yonval.o | ⊢ 𝑂 = (oppCat‘𝐶) |
yoncl.s | ⊢ 𝑆 = (SetCat‘𝑈) |
yoncl.q | ⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
yoncl.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
yoncl.h | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
Ref | Expression |
---|---|
yoncl | ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yonval.y | . . 3 ⊢ 𝑌 = (Yon‘𝐶) | |
2 | yonval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | yonval.o | . . 3 ⊢ 𝑂 = (oppCat‘𝐶) | |
4 | eqid 2818 | . . 3 ⊢ (HomF‘𝑂) = (HomF‘𝑂) | |
5 | 1, 2, 3, 4 | yonval 17499 | . 2 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF (HomF‘𝑂))) |
6 | eqid 2818 | . . 3 ⊢ (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) = (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) | |
7 | yoncl.q | . . 3 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | 3 | oppccat 16980 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 2, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | yoncl.s | . . . 4 ⊢ 𝑆 = (SetCat‘𝑈) | |
11 | yoncl.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
12 | yoncl.h | . . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
13 | 3, 4, 10, 2, 11, 12 | oppchofcl 17498 | . . 3 ⊢ (𝜑 → (HomF‘𝑂) ∈ ((𝐶 ×c 𝑂) Func 𝑆)) |
14 | 6, 7, 2, 9, 13 | curfcl 17470 | . 2 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) ∈ (𝐶 Func 𝑄)) |
15 | 5, 14 | eqeltrd 2910 | 1 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 〈cop 4563 ran crn 5549 ‘cfv 6348 (class class class)co 7145 Catccat 16923 Homf chomf 16925 oppCatcoppc 16969 Func cfunc 17112 FuncCat cfuc 17200 SetCatcsetc 17323 curryF ccurf 17448 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-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-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-homf 16929 df-comf 16930 df-oppc 16970 df-func 17116 df-nat 17201 df-fuc 17202 df-setc 17324 df-xpc 17410 df-curf 17452 df-hof 17488 df-yon 17489 |
This theorem is referenced by: yon1cl 17501 oyoncl 17508 yonedalem1 17510 yonedalem21 17511 yonedalem22 17516 yonedalem3b 17517 yonedainv 17519 yonffthlem 17520 yoniso 17523 |
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