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Theorem ressval 14558
Description: Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r  |-  R  =  ( Ws  A )
ressbas.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressval  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )

Proof of Theorem ressval
Dummy variables  a  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2  |-  R  =  ( Ws  A )
2 elex 3127 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 elex 3127 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
4 simpl 457 . . . . 5  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  W  e.  _V )
5 ovex 6320 . . . . 5  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  e.  _V
6 ifcl 3987 . . . . 5  |-  ( ( W  e.  _V  /\  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. )  e.  _V )  ->  if ( B 
C_  A ,  W ,  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )  e. 
_V )
74, 5, 6sylancl 662 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  if ( B  C_  A ,  W , 
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )  e.  _V )
8 simpl 457 . . . . . . . . 9  |-  ( ( w  =  W  /\  a  =  A )  ->  w  =  W )
98fveq2d 5876 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
10 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
119, 10syl6eqr 2526 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
12 simpr 461 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
1311, 12sseq12d 3538 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
1412, 11ineq12d 3706 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
1514opeq2d 4226 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
168, 15oveq12d 6313 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
1713, 8, 16ifbieq12d 3972 . . . . 5  |-  ( ( w  =  W  /\  a  =  A )  ->  if ( ( Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >. )
)  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
18 df-ress 14513 . . . . 5  |-s  =  ( w  e.  _V ,  a  e. 
_V  |->  if ( (
Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. ) ) )
1917, 18ovmpt2ga 6427 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V  /\  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  e. 
_V )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
207, 19mpd3an3 1325 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
212, 3, 20syl2an 477 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
221, 21syl5eq 2520 1  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ifcif 3945   <.cop 4039   ` cfv 5594  (class class class)co 6295   ndxcnx 14503   sSet csts 14504   Basecbs 14506   ↾s cress 14507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ress 14513
This theorem is referenced by:  ressid2  14559  ressval2  14560  wunress  14570
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