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Theorem rescabs2 15081
Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
rescabs2.c  |-  ( ph  ->  C  e.  V )
rescabs2.j  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
rescabs2.s  |-  ( ph  ->  S  e.  W )
rescabs2.t  |-  ( ph  ->  T  C_  S )
Assertion
Ref Expression
rescabs2  |-  ( ph  ->  ( ( Cs  S )  |`cat 
J )  =  ( C  |`cat  J ) )

Proof of Theorem rescabs2
StepHypRef Expression
1 rescabs2.s . . . 4  |-  ( ph  ->  S  e.  W )
2 rescabs2.t . . . 4  |-  ( ph  ->  T  C_  S )
3 ressabs 14570 . . . 4  |-  ( ( S  e.  W  /\  T  C_  S )  -> 
( ( Cs  S )s  T )  =  ( Cs  T ) )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  ( ( Cs  S )s  T )  =  ( Cs  T ) )
54oveq1d 6310 . 2  |-  ( ph  ->  ( ( ( Cs  S )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
6 eqid 2467 . . 3  |-  ( ( Cs  S )  |`cat  J )  =  ( ( Cs  S )  |`cat  J )
7 ovex 6320 . . . 4  |-  ( Cs  S )  e.  _V
87a1i 11 . . 3  |-  ( ph  ->  ( Cs  S )  e.  _V )
91, 2ssexd 4600 . . 3  |-  ( ph  ->  T  e.  _V )
10 rescabs2.j . . 3  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
116, 8, 9, 10rescval2 15075 . 2  |-  ( ph  ->  ( ( Cs  S )  |`cat 
J )  =  ( ( ( Cs  S )s  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
12 eqid 2467 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
13 rescabs2.c . . 3  |-  ( ph  ->  C  e.  V )
1412, 13, 9, 10rescval2 15075 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. ( Hom  `  ndx ) ,  J >. ) )
155, 11, 143eqtr4d 2518 1  |-  ( ph  ->  ( ( Cs  S )  |`cat 
J )  =  ( C  |`cat  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   <.cop 4039    X. cxp 5003    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   ndxcnx 14504   sSet csts 14505   ↾s cress 14508   Hom chom 14583    |`cat cresc 15055
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-nn 10549  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-resc 15058
This theorem is referenced by: (None)
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