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Theorem resscat 14004
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 2404 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
21ressinbas 13480 . . 3  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
32adantl 453 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
4 eqid 2404 . . . 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 2404 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
6 simpl 444 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
7 inss2 3522 . . . . . 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 14002 . . . 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 14000 . . 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 2404 . . . . . 6  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
121, 5, 6, 8, 11, 4fullresc 14003 . . . . 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 446 . . . 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 450 . . . 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 6065 . . . . 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 13890 . . 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 224 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  ( S  i^i  ( Base `  C )
) )  e.  Cat )
193, 18eqeltrd 2478 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280    X. cxp 4835    |` cres 4839   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   Catccat 13844    Homf chomf 13846  compfccomf 13847    |`cat cresc 13963
This theorem is referenced by:  ressffth  14090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-homf 13850  df-comf 13851  df-ssc 13965  df-resc 13966  df-subc 13967
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