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Theorem rescval 13956
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 2909 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 elex 2909 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
4 simpl 444 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  c  =  C )
5 simpr 448 . . . . . . . 8  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
65dmeqd 5014 . . . . . . 7  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  h  =  dom  H )
76dmeqd 5014 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  dom  h  =  dom  dom  H )
84, 7oveq12d 6040 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  ->  ( cs  dom  dom  h )  =  ( Cs  dom  dom  H ) )
95opeq2d 3935 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  -> 
<. (  Hom  `  ndx ) ,  h >.  = 
<. (  Hom  `  ndx ) ,  H >. )
108, 9oveq12d 6040 . . . 4  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( cs  dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. )  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 df-resc 13940 . . . 4  |-  |`cat  =  (
c  e.  _V ,  h  e.  _V  |->  ( ( cs 
dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. ) )
12 ovex 6047 . . . 4  |-  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
1310, 11, 12ovmpt2a 6145 . . 3  |-  ( ( C  e.  _V  /\  H  e.  _V )  ->  ( C  |`cat  H )  =  ( ( Cs  dom 
dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
142, 3, 13syl2an 464 . 2  |-  ( ( C  e.  V  /\  H  e.  W )  ->  ( C  |`cat  H )  =  ( ( Cs  dom 
dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
151, 14syl5eq 2433 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 359    = wceq 1649    e. wcel 1717   _Vcvv 2901   <.cop 3762   dom cdm 4820   ` cfv 5396  (class class class)co 6022   ndxcnx 13395   sSet csts 13396   ↾s cress 13399    Hom chom 13469    |`cat cresc 13937
This theorem is referenced by:  rescval2  13957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-resc 13940
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