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Theorem rescval 13720
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 2809 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 elex 2809 . . 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 4897 . . . . . . 7  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  h  =  dom  H )
76dmeqd 4897 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  dom  h  =  dom  dom  H )
84, 7oveq12d 5892 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  ->  ( cs  dom  dom  h )  =  ( Cs  dom  dom  H ) )
95opeq2d 3819 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  -> 
<. (  Hom  `  ndx ) ,  h >.  = 
<. (  Hom  `  ndx ) ,  H >. )
108, 9oveq12d 5892 . . . 4  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( cs  dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. )  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 df-resc 13704 . . . 4  |-  |`cat  =  (
c  e.  _V ,  h  e.  _V  |->  ( ( cs 
dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. ) )
12 ovex 5899 . . . 4  |-  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
1310, 11, 12ovmpt2a 5994 . . 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 2340 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162   ↾s cress 13165    Hom chom 13235    |`cat cresc 13701
This theorem is referenced by:  rescval2  13721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-resc 13704
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