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Theorem rescval2 15051
Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rescval.1  |-  D  =  ( C  |`cat  H )
rescval2.1  |-  ( ph  ->  C  e.  V )
rescval2.2  |-  ( ph  ->  S  e.  W )
rescval2.3  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
Assertion
Ref Expression
rescval2  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )

Proof of Theorem rescval2
StepHypRef Expression
1 rescval2.1 . . 3  |-  ( ph  ->  C  e.  V )
2 rescval2.3 . . . 4  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 rescval2.2 . . . . 5  |-  ( ph  ->  S  e.  W )
4 xpexg 6709 . . . . 5  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
53, 3, 4syl2anc 661 . . . 4  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
6 fnex 6125 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
72, 5, 6syl2anc 661 . . 3  |-  ( ph  ->  H  e.  _V )
8 rescval.1 . . . 4  |-  D  =  ( C  |`cat  H )
98rescval 15050 . . 3  |-  ( ( C  e.  V  /\  H  e.  _V )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
101, 7, 9syl2anc 661 . 2  |-  ( ph  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
11 fndm 5678 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
122, 11syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1312dmeqd 5203 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
14 dmxpid 5220 . . . . 5  |-  dom  ( S  X.  S )  =  S
1513, 14syl6eq 2524 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
1615oveq2d 6298 . . 3  |-  ( ph  ->  ( Cs  dom  dom  H )  =  ( Cs  S ) )
1716oveq1d 6297 . 2  |-  ( ph  ->  ( ( Cs  dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
1810, 17eqtrd 2508 1  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    X. cxp 4997   dom cdm 4999    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   ndxcnx 14480   sSet csts 14481   ↾s cress 14484   Hom chom 14559    |`cat cresc 15031
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 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2819  df-rex 2820  df-reu 2821  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-resc 15034
This theorem is referenced by:  rescbas  15052  reschom  15053  rescco  15055  rescabs  15056  rescabs2  15057
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