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Mirrors > Home > MPE Home > Th. List > oyoncl | Structured version Visualization version GIF version |
Description: The opposite Yoneda embedding is a functor from oppCatβπΆ to the functor category πΆ β SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Ref | Expression |
---|---|
oyoncl.o | β’ π = (oppCatβπΆ) |
oyoncl.y | β’ π = (Yonβπ) |
oyoncl.c | β’ (π β πΆ β Cat) |
oyoncl.s | β’ π = (SetCatβπ) |
oyoncl.u | β’ (π β π β π) |
oyoncl.h | β’ (π β ran (Homf βπΆ) β π) |
oyoncl.q | β’ π = (πΆ FuncCat π) |
Ref | Expression |
---|---|
oyoncl | β’ (π β π β (π Func π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oyoncl.y | . . 3 β’ π = (Yonβπ) | |
2 | oyoncl.c | . . . 4 β’ (π β πΆ β Cat) | |
3 | oyoncl.o | . . . . 5 β’ π = (oppCatβπΆ) | |
4 | 3 | oppccat 17711 | . . . 4 β’ (πΆ β Cat β π β Cat) |
5 | 2, 4 | syl 17 | . . 3 β’ (π β π β Cat) |
6 | eqid 2728 | . . 3 β’ (oppCatβπ) = (oppCatβπ) | |
7 | oyoncl.s | . . 3 β’ π = (SetCatβπ) | |
8 | eqid 2728 | . . 3 β’ ((oppCatβπ) FuncCat π) = ((oppCatβπ) FuncCat π) | |
9 | oyoncl.u | . . 3 β’ (π β π β π) | |
10 | eqid 2728 | . . . . . . 7 β’ (Homf βπΆ) = (Homf βπΆ) | |
11 | 3, 10 | oppchomf 17709 | . . . . . 6 β’ tpos (Homf βπΆ) = (Homf βπ) |
12 | 11 | rneqi 5943 | . . . . 5 β’ ran tpos (Homf βπΆ) = ran (Homf βπ) |
13 | relxp 5700 | . . . . . . 7 β’ Rel ((BaseβπΆ) Γ (BaseβπΆ)) | |
14 | eqid 2728 | . . . . . . . . . 10 β’ (BaseβπΆ) = (BaseβπΆ) | |
15 | 10, 14 | homffn 17680 | . . . . . . . . 9 β’ (Homf βπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ)) |
16 | 15 | fndmi 6663 | . . . . . . . 8 β’ dom (Homf βπΆ) = ((BaseβπΆ) Γ (BaseβπΆ)) |
17 | 16 | releqi 5783 | . . . . . . 7 β’ (Rel dom (Homf βπΆ) β Rel ((BaseβπΆ) Γ (BaseβπΆ))) |
18 | 13, 17 | mpbir 230 | . . . . . 6 β’ Rel dom (Homf βπΆ) |
19 | rntpos 8251 | . . . . . 6 β’ (Rel dom (Homf βπΆ) β ran tpos (Homf βπΆ) = ran (Homf βπΆ)) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 β’ ran tpos (Homf βπΆ) = ran (Homf βπΆ) |
21 | 12, 20 | eqtr3i 2758 | . . . 4 β’ ran (Homf βπ) = ran (Homf βπΆ) |
22 | oyoncl.h | . . . 4 β’ (π β ran (Homf βπΆ) β π) | |
23 | 21, 22 | eqsstrid 4030 | . . 3 β’ (π β ran (Homf βπ) β π) |
24 | 1, 5, 6, 7, 8, 9, 23 | yoncl 18261 | . 2 β’ (π β π β (π Func ((oppCatβπ) FuncCat π))) |
25 | oyoncl.q | . . . 4 β’ π = (πΆ FuncCat π) | |
26 | 3 | 2oppchomf 17713 | . . . . . 6 β’ (Homf βπΆ) = (Homf β(oppCatβπ)) |
27 | 26 | a1i 11 | . . . . 5 β’ (π β (Homf βπΆ) = (Homf β(oppCatβπ))) |
28 | 3 | 2oppccomf 17714 | . . . . . 6 β’ (compfβπΆ) = (compfβ(oppCatβπ)) |
29 | 28 | a1i 11 | . . . . 5 β’ (π β (compfβπΆ) = (compfβ(oppCatβπ))) |
30 | eqidd 2729 | . . . . 5 β’ (π β (Homf βπ) = (Homf βπ)) | |
31 | eqidd 2729 | . . . . 5 β’ (π β (compfβπ) = (compfβπ)) | |
32 | 6 | oppccat 17711 | . . . . . 6 β’ (π β Cat β (oppCatβπ) β Cat) |
33 | 5, 32 | syl 17 | . . . . 5 β’ (π β (oppCatβπ) β Cat) |
34 | 7 | setccat 18081 | . . . . . 6 β’ (π β π β π β Cat) |
35 | 9, 34 | syl 17 | . . . . 5 β’ (π β π β Cat) |
36 | 27, 29, 30, 31, 2, 33, 35, 35 | fucpropd 17976 | . . . 4 β’ (π β (πΆ FuncCat π) = ((oppCatβπ) FuncCat π)) |
37 | 25, 36 | eqtrid 2780 | . . 3 β’ (π β π = ((oppCatβπ) FuncCat π)) |
38 | 37 | oveq2d 7442 | . 2 β’ (π β (π Func π) = (π Func ((oppCatβπ) FuncCat π))) |
39 | 24, 38 | eleqtrrd 2832 | 1 β’ (π β π β (π Func π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3949 Γ cxp 5680 dom cdm 5682 ran crn 5683 Rel wrel 5687 βcfv 6553 (class class class)co 7426 tpos ctpos 8237 Basecbs 17187 Catccat 17651 Homf chomf 17653 compfccomf 17654 oppCatcoppc 17698 Func cfunc 17847 FuncCat cfuc 17939 SetCatcsetc 18071 Yoncyon 18248 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-hom 17264 df-cco 17265 df-cat 17655 df-cid 17656 df-homf 17657 df-comf 17658 df-oppc 17699 df-func 17851 df-nat 17940 df-fuc 17941 df-setc 18072 df-xpc 18170 df-curf 18213 df-hof 18249 df-yon 18250 |
This theorem is referenced by: oyon1cl 18270 |
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