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Theorem rescval 14019
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 2956 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 elex 2956 . . 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 5064 . . . . . . 7  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  h  =  dom  H )
76dmeqd 5064 . . . . . 6  |-  ( ( c  =  C  /\  h  =  H )  ->  dom  dom  h  =  dom  dom  H )
84, 7oveq12d 6091 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  ->  ( cs  dom  dom  h )  =  ( Cs  dom  dom  H ) )
95opeq2d 3983 . . . . 5  |-  ( ( c  =  C  /\  h  =  H )  -> 
<. (  Hom  `  ndx ) ,  h >.  = 
<. (  Hom  `  ndx ) ,  H >. )
108, 9oveq12d 6091 . . . 4  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( cs  dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. )  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 df-resc 14003 . . . 4  |-  |`cat  =  (
c  e.  _V ,  h  e.  _V  |->  ( ( cs 
dom  dom  h ) sSet  <. (  Hom  `  ndx ) ,  h >. ) )
12 ovex 6098 . . . 4  |-  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
1310, 11, 12ovmpt2a 6196 . . 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 2479 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   dom cdm 4870   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   sSet csts 13459   ↾s cress 13462    Hom chom 13532    |`cat cresc 14000
This theorem is referenced by:  rescval2  14020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-resc 14003
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