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Theorem ressval 13211
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 2809 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 elex 2809 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
4 simpl 443 . . . . 5  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  W  e.  _V )
5 ovex 5899 . . . . 5  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  e.  _V
6 ifcl 3614 . . . . 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 643 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  if ( B  C_  A ,  W , 
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )  e.  _V )
8 simpl 443 . . . . . . . . 9  |-  ( ( w  =  W  /\  a  =  A )  ->  w  =  W )
98fveq2d 5545 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
10 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
119, 10syl6eqr 2346 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
12 simpr 447 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
1311, 12sseq12d 3220 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
1412, 11ineq12d 3384 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
1514opeq2d 3819 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
168, 15oveq12d 5892 . . . . . 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 3600 . . . . 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 13171 . . . . 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 5993 . . . 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 1278 . . 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 463 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
221, 21syl5eq 2340 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ifcif 3578   <.cop 3656   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162   Basecbs 13164   ↾s cress 13165
This theorem is referenced by:  ressid2  13212  ressval2  13213  wunress  13223
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-ress 13171
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