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

Theorem resscat 15082
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 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
21ressinbas 14554 . . 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 2467 . . . 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 2467 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
6 simpl 457 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
7 inss2 3719 . . . . . 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 15080 . . . 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 15078 . . 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 2467 . . . . . 6  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
121, 5, 6, 8, 11, 4fullresc 15081 . . . . 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 6310 . . . . 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 14968 . . 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 2555 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6285   Basecbs 14493   ↾s cress 14494   Catccat 14922   Hom f chomf 14924  compfccomf 14925    |`cat cresc 15041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-homf 14928  df-comf 14929  df-ssc 15043  df-resc 15044  df-subc 15045
This theorem is referenced by:  ressffth  15168
  Copyright terms: Public domain W3C validator