MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resscat Structured version   Unicode version

Theorem resscat 15095
Description: A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
resscat  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )

Proof of Theorem resscat
StepHypRef Expression
1 eqid 2443 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
21ressinbas 14570 . . 3  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
32adantl 466 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
4 eqid 2443 . . . 4  |-  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  =  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )
5 eqid 2443 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
6 simpl 457 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
7 inss2 3704 . . . . . 6  |-  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C )
87a1i 11 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( S  i^i  ( Base `  C ) ) 
C_  ( Base `  C
) )
91, 5, 6, 8fullsubc 15093 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
) )
104, 9subccat 15091 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( C  |`cat  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  Cat )
11 eqid 2443 . . . . . 6  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
121, 5, 6, 8, 11, 4fullresc 15094 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  /\  (compf `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) ) )
1312simpld 459 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C
) ) ) )  =  ( Hom f  `  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
1412simprd 463 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) ) )
15 ovex 6309 . . . . 5  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  e.  _V
1615a1i 11 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  ( S  i^i  ( Base `  C )
) )  e.  _V )
1713, 14, 16, 10catpropd 14981 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Cs  ( S  i^i  ( Base `  C
) ) )  e. 
Cat 
<->  ( C  |`cat  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  Cat ) )
1810, 17mpbird 232 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  ( S  i^i  ( Base `  C )
) )  e.  Cat )
193, 18eqeltrd 2531 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    i^i cin 3460    C_ wss 3461    X. cxp 4987    |` cres 4991   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   Catccat 14938   Hom f chomf 14940  compfccomf 14941    |`cat cresc 15054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-hom 14598  df-cco 14599  df-cat 14942  df-cid 14943  df-homf 14944  df-comf 14945  df-ssc 15056  df-resc 15057  df-subc 15058
This theorem is referenced by:  ressffth  15181
  Copyright terms: Public domain W3C validator