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Theorem rescval2 13705
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 4800 . . . . 5  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
53, 3, 4syl2anc 642 . . . 4  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
6 fnex 5741 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
72, 5, 6syl2anc 642 . . 3  |-  ( ph  ->  H  e.  _V )
8 rescval.1 . . . 4  |-  D  =  ( C  |`cat  H )
98rescval 13704 . . 3  |-  ( ( C  e.  V  /\  H  e.  _V )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
101, 7, 9syl2anc 642 . 2  |-  ( ph  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 fndm 5343 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
122, 11syl 15 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1312dmeqd 4881 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
14 dmxpid 4898 . . . . 5  |-  dom  ( S  X.  S )  =  S
1513, 14syl6eq 2331 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
1615oveq2d 5874 . . 3  |-  ( ph  ->  ( Cs  dom  dom  H )  =  ( Cs  S ) )
1716oveq1d 5873 . 2  |-  ( ph  ->  ( ( Cs  dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
1810, 17eqtrd 2315 1  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   dom cdm 4689    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   ↾s cress 13149    Hom chom 13219    |`cat cresc 13685
This theorem is referenced by:  rescbas  13706  reschom  13707  rescco  13709  rescabs  13710  rescabs2  13711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-resc 13688
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