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Theorem ressval 13445
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 2909 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 elex 2909 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
4 simpl 444 . . . . 5  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  W  e.  _V )
5 ovex 6047 . . . . 5  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  e.  _V
6 ifcl 3720 . . . . 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 644 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  if ( B  C_  A ,  W , 
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )  e.  _V )
8 simpl 444 . . . . . . . . 9  |-  ( ( w  =  W  /\  a  =  A )  ->  w  =  W )
98fveq2d 5674 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
10 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
119, 10syl6eqr 2439 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
12 simpr 448 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
1311, 12sseq12d 3322 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
1412, 11ineq12d 3488 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
1514opeq2d 3935 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
168, 15oveq12d 6040 . . . . . 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 3706 . . . . 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 13405 . . . . 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 6144 . . . 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 1280 . . 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 464 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
221, 21syl5eq 2433 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264    C_ wss 3265   ifcif 3684   <.cop 3762   ` cfv 5396  (class class class)co 6022   ndxcnx 13395   sSet csts 13396   Basecbs 13398   ↾s cress 13399
This theorem is referenced by:  ressid2  13446  ressval2  13447  wunress  13457
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-ress 13405
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