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Theorem ressval2 13199
Description: Value of nontrivial 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
ressval2  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )

Proof of Theorem ressval2
StepHypRef Expression
1 ressbas.r . . . 4  |-  R  =  ( Ws  A )
2 ressbas.b . . . 4  |-  B  =  ( Base `  W
)
31, 2ressval 13197 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iffalse 3574 . . 3  |-  ( -.  B  C_  A  ->  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
53, 4sylan9eqr 2339 . 2  |-  ( ( -.  B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )
653impb 1147 1  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   ifcif 3567   <.cop 3645   ` cfv 5257  (class class class)co 5860   ndxcnx 13147   sSet csts 13148   Basecbs 13150   ↾s cress 13151
This theorem is referenced by:  ressbas  13200  resslem  13203  ressinbas  13206  ressress  13207  rescabs  13712  mgpress  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-ress 13157
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