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Theorem rescval2 14030
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 4991 . . . . 5  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
53, 3, 4syl2anc 644 . . . 4  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
6 fnex 5963 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
72, 5, 6syl2anc 644 . . 3  |-  ( ph  ->  H  e.  _V )
8 rescval.1 . . . 4  |-  D  =  ( C  |`cat  H )
98rescval 14029 . . 3  |-  ( ( C  e.  V  /\  H  e.  _V )  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
101, 7, 9syl2anc 644 . 2  |-  ( ph  ->  D  =  ( ( Cs 
dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
11 fndm 5546 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
122, 11syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1312dmeqd 5074 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
14 dmxpid 5091 . . . . 5  |-  dom  ( S  X.  S )  =  S
1513, 14syl6eq 2486 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
1615oveq2d 6099 . . 3  |-  ( ph  ->  ( Cs  dom  dom  H )  =  ( Cs  S ) )
1716oveq1d 6098 . 2  |-  ( ph  ->  ( ( Cs  dom  dom  H ) sSet  <. (  Hom  `  ndx ) ,  H >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
1810, 17eqtrd 2470 1  |-  ( ph  ->  D  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    X. cxp 4878   dom cdm 4880    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   ndxcnx 13468   sSet csts 13469   ↾s cress 13472    Hom chom 13542    |`cat cresc 14010
This theorem is referenced by:  rescbas  14031  reschom  14032  rescco  14034  rescabs  14035  rescabs2  14036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-resc 14013
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