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Theorem rescval2 14741
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 6507 . . . . 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 5944 . . . 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 14740 . . 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 5510 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
122, 11syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1312dmeqd 5042 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
14 dmxpid 5059 . . . . 5  |-  dom  ( S  X.  S )  =  S
1513, 14syl6eq 2491 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
1615oveq2d 6107 . . 3  |-  ( ph  ->  ( Cs  dom  dom  H )  =  ( Cs  S ) )
1716oveq1d 6106 . 2  |-  ( ph  ->  ( ( Cs  dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
1810, 17eqtrd 2475 1  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2972   <.cop 3883    X. cxp 4838   dom cdm 4840    Fn wfn 5413   ` cfv 5418  (class class class)co 6091   ndxcnx 14171   sSet csts 14172   ↾s cress 14175   Hom chom 14249    |`cat cresc 14721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-resc 14724
This theorem is referenced by:  rescbas  14742  reschom  14743  rescco  14745  rescabs  14746  rescabs2  14747
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