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Mirrors > Home > MPE Home > Th. List > yoniso | Structured version Visualization version Unicode version |
Description: If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.) |
Ref | Expression |
---|---|
yoniso.y | Yon |
yoniso.o | oppCat |
yoniso.s | |
yoniso.d | CatCat |
yoniso.b | |
yoniso.i | |
yoniso.q | FuncCat |
yoniso.e | ↾s |
yoniso.v | |
yoniso.c | |
yoniso.u | |
yoniso.h | f |
yoniso.eb | |
yoniso.1 |
Ref | Expression |
---|---|
yoniso |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfunc 16522 | . . . 4 | |
2 | yoniso.y | . . . . 5 Yon | |
3 | yoniso.d | . . . . . . . 8 CatCat | |
4 | yoniso.b | . . . . . . . 8 | |
5 | yoniso.v | . . . . . . . 8 | |
6 | 3, 4, 5 | catcbas 16747 | . . . . . . 7 |
7 | inss2 3834 | . . . . . . 7 | |
8 | 6, 7 | syl6eqss 3655 | . . . . . 6 |
9 | yoniso.c | . . . . . 6 | |
10 | 8, 9 | sseldd 3604 | . . . . 5 |
11 | yoniso.o | . . . . 5 oppCat | |
12 | yoniso.s | . . . . 5 | |
13 | yoniso.q | . . . . 5 FuncCat | |
14 | yoniso.u | . . . . 5 | |
15 | yoniso.h | . . . . 5 f | |
16 | 2, 10, 11, 12, 13, 14, 15 | yoncl 16902 | . . . 4 |
17 | 1st2nd 7214 | . . . 4 | |
18 | 1, 16, 17 | sylancr 695 | . . 3 |
19 | 2, 11, 12, 13, 10, 14, 15 | yonffth 16924 | . . . . 5 Full Faith |
20 | 18, 19 | eqeltrrd 2702 | . . . 4 Full Faith |
21 | eqid 2622 | . . . . . 6 | |
22 | yoniso.e | . . . . . 6 ↾s | |
23 | 11 | oppccat 16382 | . . . . . . . 8 |
24 | 10, 23 | syl 17 | . . . . . . 7 |
25 | 12 | setccat 16735 | . . . . . . . 8 |
26 | 14, 25 | syl 17 | . . . . . . 7 |
27 | 13, 24, 26 | fuccat 16630 | . . . . . 6 |
28 | fvex 6201 | . . . . . . . 8 | |
29 | 28 | rnex 7100 | . . . . . . 7 |
30 | 29 | a1i 11 | . . . . . 6 |
31 | 13 | fucbas 16620 | . . . . . . . . 9 |
32 | 1st2ndbr 7217 | . . . . . . . . . 10 | |
33 | 1, 16, 32 | sylancr 695 | . . . . . . . . 9 |
34 | 21, 31, 33 | funcf1 16526 | . . . . . . . 8 |
35 | ffn 6045 | . . . . . . . 8 | |
36 | 34, 35 | syl 17 | . . . . . . 7 |
37 | dffn3 6054 | . . . . . . 7 | |
38 | 36, 37 | sylib 208 | . . . . . 6 |
39 | 21, 22, 27, 30, 38 | ffthres2c 16600 | . . . . 5 Full Faith Full Faith |
40 | df-br 4654 | . . . . 5 Full Faith Full Faith | |
41 | df-br 4654 | . . . . 5 Full Faith Full Faith | |
42 | 39, 40, 41 | 3bitr3g 302 | . . . 4 Full Faith Full Faith |
43 | 20, 42 | mpbid 222 | . . 3 Full Faith |
44 | 18, 43 | eqeltrd 2701 | . 2 Full Faith |
45 | fveq2 6191 | . . . . . . . . 9 | |
46 | 45 | fveq1d 6193 | . . . . . . . 8 |
47 | 46 | fveq2d 6195 | . . . . . . 7 |
48 | simpl 473 | . . . . . . . . . 10 | |
49 | 48, 48 | jca 554 | . . . . . . . . 9 |
50 | eleq1 2689 | . . . . . . . . . . . . 13 | |
51 | 50 | anbi2d 740 | . . . . . . . . . . . 12 |
52 | 51 | anbi2d 740 | . . . . . . . . . . 11 |
53 | fveq2 6191 | . . . . . . . . . . . . . . 15 | |
54 | 53 | fveq2d 6195 | . . . . . . . . . . . . . 14 |
55 | 54 | fveq1d 6193 | . . . . . . . . . . . . 13 |
56 | 55 | fveq2d 6195 | . . . . . . . . . . . 12 |
57 | id 22 | . . . . . . . . . . . 12 | |
58 | 56, 57 | eqeq12d 2637 | . . . . . . . . . . 11 |
59 | 52, 58 | imbi12d 334 | . . . . . . . . . 10 |
60 | 10 | adantr 481 | . . . . . . . . . . . . 13 |
61 | simprr 796 | . . . . . . . . . . . . 13 | |
62 | eqid 2622 | . . . . . . . . . . . . 13 | |
63 | simprl 794 | . . . . . . . . . . . . 13 | |
64 | 2, 21, 60, 61, 62, 63 | yon11 16904 | . . . . . . . . . . . 12 |
65 | 64 | fveq2d 6195 | . . . . . . . . . . 11 |
66 | yoniso.1 | . . . . . . . . . . 11 | |
67 | 65, 66 | eqtrd 2656 | . . . . . . . . . 10 |
68 | 59, 67 | chvarv 2263 | . . . . . . . . 9 |
69 | 49, 68 | sylan2 491 | . . . . . . . 8 |
70 | 69, 67 | eqeq12d 2637 | . . . . . . 7 |
71 | 47, 70 | syl5ib 234 | . . . . . 6 |
72 | 71 | ralrimivva 2971 | . . . . 5 |
73 | dff13 6512 | . . . . 5 | |
74 | 34, 72, 73 | sylanbrc 698 | . . . 4 |
75 | f1f1orn 6148 | . . . 4 | |
76 | 74, 75 | syl 17 | . . 3 |
77 | frn 6053 | . . . . . 6 | |
78 | 34, 77 | syl 17 | . . . . 5 |
79 | 22, 31 | ressbas2 15931 | . . . . 5 |
80 | 78, 79 | syl 17 | . . . 4 |
81 | f1oeq3 6129 | . . . 4 | |
82 | 80, 81 | syl 17 | . . 3 |
83 | 76, 82 | mpbid 222 | . 2 |
84 | eqid 2622 | . . 3 | |
85 | yoniso.eb | . . 3 | |
86 | yoniso.i | . . 3 | |
87 | 3, 4, 21, 84, 5, 9, 85, 86 | catciso 16757 | . 2 Full Faith |
88 | 44, 83, 87 | mpbir2and 957 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cvv 3200 cin 3573 wss 3574 cop 4183 class class class wbr 4653 crn 5115 wrel 5119 wfn 5883 wf 5884 wf1 5885 wf1o 5887 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 cbs 15857 ↾s cress 15858 chom 15952 ccat 16325 f chomf 16327 oppCatcoppc 16371 ciso 16406 cfunc 16514 Full cful 16562 Faith cfth 16563 FuncCat cfuc 16602 csetc 16725 CatCatccatc 16744 Yoncyon 16889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-hom 15966 df-cco 15967 df-cat 16329 df-cid 16330 df-homf 16331 df-comf 16332 df-oppc 16372 df-sect 16407 df-inv 16408 df-iso 16409 df-ssc 16470 df-resc 16471 df-subc 16472 df-func 16518 df-idfu 16519 df-cofu 16520 df-full 16564 df-fth 16565 df-nat 16603 df-fuc 16604 df-setc 16726 df-catc 16745 df-xpc 16812 df-1stf 16813 df-2ndf 16814 df-prf 16815 df-evlf 16853 df-curf 16854 df-hof 16890 df-yon 16891 |
This theorem is referenced by: (None) |
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