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Theorem rescval 13704
Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
rescval.1  |-  D  =  ( C  |`cat  H )
Assertion
Ref Expression
rescval  |-  ( ( C  e.  V  /\  H  e.  W )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )

Proof of Theorem rescval
Dummy variables  h  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rescval.1 . 2  |-  D  =  ( C  |`cat  H )
2 elex 2796 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 elex 2796 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
4 simpl 443 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  c  =  C )
5 simpr 447 . . . . . . . 8  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
65dmeqd 4881 . . . . . . 7  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  h  =  dom  H )
76dmeqd 4881 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  dom  h  =  dom  dom  H )
84, 7oveq12d 5876 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  ->  ( cs  dom  dom  h )  =  ( Cs  dom  dom  H ) )
95opeq2d 3803 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  -> 
<. (  Hom  `  ndx ) ,  h >.  = 
<. (  Hom  `  ndx ) ,  H >. )
108, 9oveq12d 5876 . . . 4  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( cs  dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. )  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 df-resc 13688 . . . 4  |-  |`cat  =  (
c  e.  _V ,  h  e.  _V  |->  ( ( cs 
dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. ) )
12 ovex 5883 . . . 4  |-  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
1310, 11, 12ovmpt2a 5978 . . 3  |-  ( ( C  e.  _V  /\  H  e.  _V )  ->  ( C  |`cat  H )  =  ( ( Cs  dom 
dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
142, 3, 13syl2an 463 . 2  |-  ( ( C  e.  V  /\  H  e.  W )  ->  ( C  |`cat  H )  =  ( ( Cs  dom 
dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
151, 14syl5eq 2327 1  |-  ( ( C  e.  V  /\  H  e.  W )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   dom cdm 4689   ` 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:  rescval2  13705
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-resc 13688
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