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Theorem rescval2 15441
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 6584 . . . . 5  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
53, 3, 4syl2anc 659 . . . 4  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
6 fnex 6120 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
72, 5, 6syl2anc 659 . . 3  |-  ( ph  ->  H  e.  _V )
8 rescval.1 . . . 4  |-  D  =  ( C  |`cat  H )
98rescval 15440 . . 3  |-  ( ( C  e.  V  /\  H  e.  _V )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
101, 7, 9syl2anc 659 . 2  |-  ( ph  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
11 fndm 5661 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
122, 11syl 17 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1312dmeqd 5026 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
14 dmxpid 5043 . . . . 5  |-  dom  ( S  X.  S )  =  S
1513, 14syl6eq 2459 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
1615oveq2d 6294 . . 3  |-  ( ph  ->  ( Cs  dom  dom  H )  =  ( Cs  S ) )
1716oveq1d 6293 . 2  |-  ( ph  ->  ( ( Cs  dom  dom  H ) sSet  <. ( Hom  `  ndx ) ,  H >. )  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
1810, 17eqtrd 2443 1  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059   <.cop 3978    X. cxp 4821   dom cdm 4823    Fn wfn 5564   ` cfv 5569  (class class class)co 6278   ndxcnx 14838   sSet csts 14839   ↾s cress 14842   Hom chom 14920    |`cat cresc 15421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-resc 15424
This theorem is referenced by:  rescbas  15442  reschom  15443  rescco  15445  rescabs  15446  rescabs2  15447  dfrngc2  38291  dfringc2  38337  rngcresringcat  38349  rngcrescrhm  38404  rngcrescrhmALTV  38423
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