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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 2736 | . . 3 ⊢ (HomF‘𝑂) = (HomF‘𝑂) | |
5 | 1, 2, 3, 4 | yonval 17723 | . 2 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF (HomF‘𝑂))) |
6 | eqid 2736 | . . 3 ⊢ (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) = (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) | |
7 | yoncl.q | . . 3 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
8 | 3 | oppccat 17180 | . . . 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 17722 | . . 3 ⊢ (𝜑 → (HomF‘𝑂) ∈ ((𝐶 ×c 𝑂) Func 𝑆)) |
14 | 6, 7, 2, 9, 13 | curfcl 17694 | . 2 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF (HomF‘𝑂)) ∈ (𝐶 Func 𝑄)) |
15 | 5, 14 | eqeltrd 2831 | 1 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 〈cop 4533 ran crn 5537 ‘cfv 6358 (class class class)co 7191 Catccat 17121 Homf chomf 17123 oppCatcoppc 17168 Func cfunc 17314 FuncCat cfuc 17403 SetCatcsetc 17535 curryF ccurf 17672 HomFchof 17710 Yoncyon 17711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-hom 16773 df-cco 16774 df-cat 17125 df-cid 17126 df-homf 17127 df-comf 17128 df-oppc 17169 df-func 17318 df-nat 17404 df-fuc 17405 df-setc 17536 df-xpc 17633 df-curf 17676 df-hof 17712 df-yon 17713 |
This theorem is referenced by: yon1cl 17725 oyoncl 17732 yonedalem1 17734 yonedalem21 17735 yonedalem22 17740 yonedalem3b 17741 yonedainv 17743 yonffthlem 17744 yoniso 17747 |
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