Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > funcres | Structured version Visualization version Unicode version | ||
| Description: A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| Ref | Expression |
|---|---|
| funcres.f |
          |
| funcres.h |
 Subcat |
| Ref | Expression |
|---|---|
| funcres |
 f cat
|
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funcres.f |
. . . 4
| |
| 2 | funcres.h |
. . . 4
Subcat | |
| 3 | 1, 2 | resfval 16552 |
. . 3
f         |
| 4 | 3 | fveq2d 6195 |
. . . . 5
f |
| 5 | fvex 6201 |
. . . . . . 7
 | |
| 6 | 5 | resex 5443 |
. . . . . 6
|
| 7 | dmexg 7097 |
. . . . . . 7
Subcat
| |
| 8 | mptexg 6484 |
. . . . . . 7
| |
| 9 | 2, 7, 8 | 3syl 18 |
. . . . . 6
|
| 10 | op2ndg 7181 |
. . . . . 6
![/\](wedge.gif "/\"){kind=small} | |
| 11 | 6, 9, 10 | sylancr 695 |
. . . . 5
|
| 12 | 4, 11 | eqtrd 2656 |
. . . 4
f |
| 13 | 12 | opeq2d 4409 |
. . 3
f |
| 14 | 3, 13 | eqtr4d 2659 |
. 2
f f |
| 15 | eqid 2622 |
. . . 4
 cat
cat
| |
| 16 | eqid 2622 |
. . . 4
| |
| 17 | eqid 2622 |
. . . 4
 cat
cat
| |
| 18 | eqid 2622 |
. . . 4
| |
| 19 | eqid 2622 |
. . . 4
 cat cat | |
| 20 | eqid 2622 |
. . . 4
| |
| 21 | eqid 2622 |
. . . 4
comp
cat comp cat | |
| 22 | eqid 2622 |
. . . 4
comp comp | |
| 23 | eqid 2622 |
. . . . 5
cat cat | |
| 24 | 23, 2 | subccat 16508 |
. . . 4
cat
 |
| 25 | funcrcl 16523 |
. . . . . 6
| |
| 26 | 1, 25 | syl 17 |
. . . . 5
|
| 27 | 26 | simprd 479 |
. . . 4
|
| 28 | eqid 2622 |
. . . . . . 7
| |
| 29 | relfunc 16522 |
. . . . . . . 8
 | |
| 30 | 1st2ndbr 7217 |
. . . . . . . 8
| |
| 31 | 29, 1, 30 | sylancr 695 |
. . . . . . 7
|
| 32 | 28, 16, 31 | funcf1 16526 |
. . . . . 6
  |
| 33 | eqidd 2623 |
. . . . . . . 8
| |
| 34 | 2, 33 | subcfn 16501 |
. . . . . . 7
 |
| 35 | 2, 34, 28 | subcss1 16502 |
. . . . . 6
 |
| 36 | 32, 35 | fssresd 6071 |
. . . . 5
|
| 37 | 26 | simpld 475 |
. . . . . . 7
|
| 38 | 23, 28, 37, 34, 35 | rescbas 16489 |
. . . . . 6
cat |
| 39 | 38 | feq2d 6031 |
. . . . 5
 cat
|
| 40 | 36, 39 | mpbid 222 |
. . . 4
cat
|
| 41 | fvex 6201 |
. . . . . . 7
| |
| 42 | 41 | resex 5443 |
. . . . . 6
|
| 43 | eqid 2622 |
. . . . . 6
| |
| 44 | 42, 43 | fnmpti 6022 |
. . . . 5
|
| 45 | 12 | eqcomd 2628 |
. . . . . 6
f |
| 46 | fndm 5990 |
. . . . . . . 8
| |
| 47 | 34, 46 | syl 17 |
. . . . . . 7
|
| 48 | 38 | sqxpeqd 5141 |
. . . . . . 7
cat
cat
|
| 49 | 47, 48 | eqtrd 2656 |
. . . . . 6
cat cat |
| 50 | 45, 49 | fneq12d 5983 |
. . . . 5
f cat cat |
| 51 | 44, 50 | mpbii 223 |
. . . 4
f cat cat |
| 52 | eqid 2622 |
. . . . . . . 8
| |
| 53 | 31 | adantr 481 |
. . . . . . . 8
cat
cat
|
| 54 | 35 | adantr 481 |
. . . . . . . . 9
cat
cat
|
| 55 | simprl 794 |
. . . . . . . . . 10
cat
cat
cat
| |
| 56 | 38 | adantr 481 |
. . . . . . . . . 10
cat
cat
cat |
| 57 | 55, 56 | eleqtrrd 2704 |
. . . . . . . . 9
cat
cat
|
| 58 | 54, 57 | sseldd 3604 |
. . . . . . . 8
cat
cat
|
| 59 | simprr 796 |
. . . . . . . . . 10
cat
cat
cat
| |
| 60 | 59, 56 | eleqtrrd 2704 |
. . . . . . . . 9
cat
cat
|
| 61 | 54, 60 | sseldd 3604 |
. . . . . . . 8
cat
cat
|
| 62 | 28, 52, 18, 53, 58, 61 | funcf2 16528 |
. . . . . . 7
cat
cat
|
| 63 | 2 | adantr 481 |
. . . . . . . 8
cat
cat
Subcat |
| 64 | 34 | adantr 481 |
. . . . . . . 8
cat
cat
|
| 65 | 63, 64, 52, 57, 60 | subcss2 16503 |
. . . . . . 7
cat
cat
|
| 66 | 62, 65 | fssresd 6071 |
. . . . . 6
cat
cat
|
| 67 | 1 | adantr 481 |
. . . . . . . 8
cat
cat
|
| 68 | 67, 63, 64, 57, 60 | resf2nd 16555 |
. . . . . . 7
cat
cat
f |
| 69 | 68 | feq1d 6030 |
. . . . . 6
cat
cat
f
|
| 70 | 66, 69 | mpbird 247 |
. . . . 5
cat
cat
f
|
| 71 | 23, 28, 37, 34, 35 | reschom 16490 |
. . . . . . . 8
cat
|
| 72 | 71 | adantr 481 |
. . . . . . 7
cat
cat
cat
|
| 73 | 72 | oveqd 6667 |
. . . . . 6
cat
cat
cat |
| 74 | fvres 6207 |
. . . . . . . . 9
| |
| 75 | 57, 74 | syl 17 |
. . . . . . . 8
cat
cat
|
| 76 | fvres 6207 |
. . . . . . . . 9
| |
| 77 | 60, 76 | syl 17 |
. . . . . . . 8
cat
cat
|
| 78 | 75, 77 | oveq12d 6668 |
. . . . . . 7
cat
cat
|
| 79 | 78 | eqcomd 2628 |
. . . . . 6
cat
cat
|
| 80 | 73, 79 | feq23d 6040 |
. . . . 5
cat
cat
f
f cat
|
| 81 | 70, 80 | mpbid 222 |
. . . 4
cat
cat
f cat
|
| 82 | 1 | adantr 481 |
. . . . . . 7
cat
|
| 83 | 2 | adantr 481 |
. . . . . . 7
cat
Subcat |
| 84 | 34 | adantr 481 |
. . . . . . 7
cat
|
| 85 | 38 | eleq2d 2687 |
. . . . . . . 8
cat |
| 86 | 85 | biimpar 502 |
. . . . . . 7
cat
|
| 87 | 82, 83, 84, 86, 86 | resf2nd 16555 |
. . . . . 6
cat f |
| 88 | eqid 2622 |
. . . . . . . 8
| |
| 89 | 23, 83, 84, 88, 86 | subcid 16507 |
. . . . . . 7
cat cat |
| 90 | 89 | eqcomd 2628 |
. . . . . 6
cat
cat |
| 91 | 87, 90 | fveq12d 6197 |
. . . . 5
cat f cat |
| 92 | 31 | adantr 481 |
. . . . . . 7
cat |
| 93 | 38, 35 | eqsstr3d 3640 |
. . . . . . . 8
cat |
| 94 | 93 | sselda 3603 |
. . . . . . 7
cat
|
| 95 | 28, 88, 20, 92, 94 | funcid 16530 |
. . . . . 6
cat |
| 96 | 83, 84, 86, 88 | subcidcl 16504 |
. . . . . . 7
cat |
| 97 | fvres 6207 |
. . . . . . 7
| |
| 98 | 96, 97 | syl 17 |
. . . . . 6
cat |
| 99 | 86, 74 | syl 17 |
. . . . . . 7
cat |
| 100 | 99 | fveq2d 6195 |
. . . . . 6
cat |
| 101 | 95, 98, 100 | 3eqtr4d 2666 |
. . . . 5
cat
|
| 102 | 91, 101 | eqtrd 2656 |
. . . 4
cat f cat |
| 103 | 2 | 3ad2ant1 1082 |
. . . . . . . 8
cat
cat
cat cat cat
Subcat |
| 104 | 34 | 3ad2ant1 1082 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 105 | simp21 1094 |
. . . . . . . . 9
cat
cat
cat cat cat
cat | |
| 106 | 38 | 3ad2ant1 1082 |
. . . . . . . . 9
cat
cat
cat cat cat
cat |
| 107 | 105, 106 | eleqtrrd 2704 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 108 | eqid 2622 |
. . . . . . . 8
comp comp | |
| 109 | simp22 1095 |
. . . . . . . . 9
cat
cat
cat cat cat
cat | |
| 110 | 109, 106 | eleqtrrd 2704 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 111 | simp23 1096 |
. . . . . . . . 9
cat
cat
cat cat cat
cat | |
| 112 | 111, 106 | eleqtrrd 2704 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 113 | simp3l 1089 |
. . . . . . . . 9
cat
cat
cat cat cat
cat
| |
| 114 | 71 | 3ad2ant1 1082 |
. . . . . . . . . 10
cat
cat
cat cat cat
cat |
| 115 | 114 | oveqd 6667 |
. . . . . . . . 9
cat
cat
cat cat cat
cat
|
| 116 | 113, 115 | eleqtrrd 2704 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 117 | simp3r 1090 |
. . . . . . . . 9
cat
cat
cat cat cat
cat
| |
| 118 | 114 | oveqd 6667 |
. . . . . . . . 9
cat
cat
cat cat cat
cat
|
| 119 | 117, 118 | eleqtrrd 2704 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 120 | 103, 104, 107, 108, 110, 112, 116, 119 | subccocl 16505 |
. . . . . . 7
cat
cat
cat cat cat
comp |
| 121 | fvres 6207 |
. . . . . . 7
comp comp comp | |
| 122 | 120, 121 | syl 17 |
. . . . . 6
cat
cat
cat cat cat
comp comp |
| 123 | 31 | 3ad2ant1 1082 |
. . . . . . 7
cat
cat
cat cat cat
|
| 124 | 35 | 3ad2ant1 1082 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 125 | 124, 107 | sseldd 3604 |
. . . . . . 7
cat
cat
cat cat cat
|
| 126 | 124, 110 | sseldd 3604 |
. . . . . . 7
cat
cat
cat cat cat
|
| 127 | 124, 112 | sseldd 3604 |
. . . . . . 7
cat
cat
cat cat cat
|
| 128 | 103, 104, 52, 107, 110 | subcss2 16503 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 129 | 128, 116 | sseldd 3604 |
. . . . . . 7
cat
cat
cat cat cat
|
| 130 | 103, 104, 52, 110, 112 | subcss2 16503 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 131 | 130, 119 | sseldd 3604 |
. . . . . . 7
cat
cat
cat cat cat
|
| 132 | 28, 52, 108, 22, 123, 125, 126, 127, 129, 131 | funcco 16531 |
. . . . . 6
cat
cat
cat cat cat
comp comp |
| 133 | 122, 132 | eqtrd 2656 |
. . . . 5
cat
cat
cat cat cat
comp comp |
| 134 | 1 | 3ad2ant1 1082 |
. . . . . . 7
cat
cat
cat cat cat
|
| 135 | 134, 103, 104, 107, 112 | resf2nd 16555 |
. . . . . 6
cat
cat
cat cat cat
f |
| 136 | 23, 28, 37, 34, 35, 108 | rescco 16492 |
. . . . . . . . . 10
comp comp cat |
| 137 | 136 | 3ad2ant1 1082 |
. . . . . . . . 9
cat
cat
cat cat cat
comp comp
cat |
| 138 | 137 | eqcomd 2628 |
. . . . . . . 8
cat
cat
cat cat cat
comp cat comp |
| 139 | 138 | oveqd 6667 |
. . . . . . 7
cat
cat
cat cat cat
comp cat
comp |
| 140 | 139 | oveqd 6667 |
. . . . . 6
cat
cat
cat cat cat
comp cat comp |
| 141 | 135, 140 | fveq12d 6197 |
. . . . 5
cat
cat
cat cat cat
f comp cat comp |
| 142 | 107, 74 | syl 17 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 143 | 110, 76 | syl 17 |
. . . . . . . 8
cat
cat
cat cat cat
|
| 144 | 142, 143 | opeq12d 4410 |
. . . . . . 7
cat
cat
cat cat cat
|
| 145 | fvres 6207 |
. . . . . . . 8
| |
| 146 | 112, 145 | syl 17 |
. . . . . . 7
cat
cat
cat cat cat
|
| 147 | 144, 146 | oveq12d 6668 |
. . . . . 6
cat
cat
cat cat cat
comp comp |
| 148 | 134, 103, 104, 110, 112 | resf2nd 16555 |
. . . . . . . 8
cat
cat
cat cat cat
f |
| 149 | 148 | fveq1d 6193 |
. . . . . . 7
cat
cat
cat cat cat
f |
| 150 | fvres 6207 |
. . . . . . . 8
| |
| 151 | 119, 150 | syl 17 |
. . . . . . 7
cat
cat
cat cat cat
|
| 152 | 149, 151 | eqtrd 2656 |
. . . . . 6
cat
cat
cat cat cat
f |
| 153 | 134, 103, 104, 107, 110 | resf2nd 16555 |
. . . . . . . 8
cat
cat
cat cat cat
f |
| 154 | 153 | fveq1d 6193 |
. . . . . . 7
cat
cat
cat cat cat
f |
| 155 | fvres 6207 |
. . . . . . . 8
| |
| 156 | 116, 155 | syl 17 |
. . . . . . 7
cat
cat
cat cat cat
|
| 157 | 154, 156 | eqtrd 2656 |
. . . . . 6
cat
cat
cat cat cat
f |
| 158 | 147, 152, 157 | oveq123d 6671 |
. . . . 5
cat
cat
cat cat cat
f
comp f comp |
| 159 | 133, 141, 158 | 3eqtr4d 2666 |
. . . 4
cat
cat
cat cat cat
f comp cat f
comp f |
| 160 | 15, 16, 17, 18, 19, 20, 21, 22, 24, 27, 40, 51, 81, 102, 159 | isfuncd 16525 |
. . 3
cat f |
| 161 | df-br 4654 |
. . 3
cat f
f cat
| |
| 162 | 160, 161 | sylib 208 |
. 2
f cat
|
| 163 | 14, 162 | eqeltrd 2701 |
1
f cat
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wa 384
w3a 1037
wceq 1483
wcel 1990
cvv 3200
wss 3574
cop 4183
class class class wbr 4653 cmpt 4729 cxp 5112 cdm 5114 cres 5116 wrel 5119 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650
c1st 7166
c2nd 7167
cbs 15857
chom 15952
compcco 15953 ccat 16325 ccid 16326 cat cresc 16468 Subcatcsubc 16469 cfunc 16514 f cresf 16517 |
| 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-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-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 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-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-ssc 16470 df-resc 16471 df-subc 16472 df-func 16518 df-resf 16521 |
| This theorem is referenced by: funcrngcsetc 41998 funcringcsetc 42035 |
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