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Theorem resscat 14054
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 2438 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
21ressinbas 13530 . . 3  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
32adantl 454 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
4 eqid 2438 . . . 4  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  =  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )
5 eqid 2438 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
6 simpl 445 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
7 inss2 3564 . . . . . 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 14052 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
) )
104, 9subccat 14050 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  Cat )
11 eqid 2438 . . . . . 6  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
121, 5, 6, 8, 11, 4fullresc 14053 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( (  Homf  `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (  Homf  `  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  /\  (compf `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) ) )
1312simpld 447 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  ( Cs  ( S  i^i  ( Base `  C
) ) ) )  =  (  Homf  `  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
1412simprd 451 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) ) )
15 ovex 6109 . . . . 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 13940 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Cs  ( S  i^i  ( Base `  C
) ) )  e. 
Cat 
<->  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  Cat ) )
1810, 17mpbird 225 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  ( S  i^i  ( Base `  C )
) )  e.  Cat )
193, 18eqeltrd 2512 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322    X. cxp 4879    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475   Catccat 13894    Homf chomf 13896  compfccomf 13897    |`cat cresc 14013
This theorem is referenced by:  ressffth  14140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-homf 13900  df-comf 13901  df-ssc 14015  df-resc 14016  df-subc 14017
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