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Theorem ressval2 14227
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 14225 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iffalse 3799 . . 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 2497 . 2  |-  ( ( -.  B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )
653impb 1183 1  |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y
)  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3327    C_ wss 3328   ifcif 3791   <.cop 3883   ` cfv 5418  (class class class)co 6091   ndxcnx 14171   sSet csts 14172   Basecbs 14174   ↾s cress 14175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-ress 14181
This theorem is referenced by:  ressbas  14228  resslem  14231  ressinbas  14234  ressress  14235  rescabs  14746  mgpress  16602
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