![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > yon1cl | Structured version Visualization version GIF version |
Description: The Yoneda embedding at an object of πΆ is a presheaf on πΆ, also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.) |
Ref | Expression |
---|---|
yon11.y | β’ π = (YonβπΆ) |
yon11.b | β’ π΅ = (BaseβπΆ) |
yon11.c | β’ (π β πΆ β Cat) |
yon11.p | β’ (π β π β π΅) |
yon1cl.o | β’ π = (oppCatβπΆ) |
yon1cl.s | β’ π = (SetCatβπ) |
yon1cl.u | β’ (π β π β π) |
yon1cl.h | β’ (π β ran (Homf βπΆ) β π) |
Ref | Expression |
---|---|
yon1cl | β’ (π β ((1st βπ)βπ) β (π Func π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yon11.b | . . 3 β’ π΅ = (BaseβπΆ) | |
2 | eqid 2724 | . . . 4 β’ (π FuncCat π) = (π FuncCat π) | |
3 | 2 | fucbas 17920 | . . 3 β’ (π Func π) = (Baseβ(π FuncCat π)) |
4 | relfunc 17817 | . . . 4 β’ Rel (πΆ Func (π FuncCat π)) | |
5 | yon11.y | . . . . 5 β’ π = (YonβπΆ) | |
6 | yon11.c | . . . . 5 β’ (π β πΆ β Cat) | |
7 | yon1cl.o | . . . . 5 β’ π = (oppCatβπΆ) | |
8 | yon1cl.s | . . . . 5 β’ π = (SetCatβπ) | |
9 | yon1cl.u | . . . . 5 β’ (π β π β π) | |
10 | yon1cl.h | . . . . 5 β’ (π β ran (Homf βπΆ) β π) | |
11 | 5, 6, 7, 8, 2, 9, 10 | yoncl 18223 | . . . 4 β’ (π β π β (πΆ Func (π FuncCat π))) |
12 | 1st2ndbr 8022 | . . . 4 β’ ((Rel (πΆ Func (π FuncCat π)) β§ π β (πΆ Func (π FuncCat π))) β (1st βπ)(πΆ Func (π FuncCat π))(2nd βπ)) | |
13 | 4, 11, 12 | sylancr 586 | . . 3 β’ (π β (1st βπ)(πΆ Func (π FuncCat π))(2nd βπ)) |
14 | 1, 3, 13 | funcf1 17821 | . 2 β’ (π β (1st βπ):π΅βΆ(π Func π)) |
15 | yon11.p | . 2 β’ (π β π β π΅) | |
16 | 14, 15 | ffvelcdmd 7078 | 1 β’ (π β ((1st βπ)βπ) β (π Func π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3941 class class class wbr 5139 ran crn 5668 Rel wrel 5672 βcfv 6534 (class class class)co 7402 1st c1st 7967 2nd c2nd 7968 Basecbs 17149 Catccat 17613 Homf chomf 17615 oppCatcoppc 17660 Func cfunc 17809 FuncCat cfuc 17901 SetCatcsetc 18033 Yoncyon 18210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-homf 17619 df-comf 17620 df-oppc 17661 df-func 17813 df-nat 17902 df-fuc 17903 df-setc 18034 df-xpc 18132 df-curf 18175 df-hof 18211 df-yon 18212 |
This theorem is referenced by: yonedalem21 18234 yonedalem3a 18235 yonedalem4c 18238 |
Copyright terms: Public domain | W3C validator |