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Theorem ressval2 13510
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 13508 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iffalse 3738 . . 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 2489 . 2  |-  ( ( -.  B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. ) )
653impb 1149 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   ifcif 3731   <.cop 3809   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   sSet csts 13459   Basecbs 13461   ↾s cress 13462
This theorem is referenced by:  ressbas  13511  resslem  13514  ressinbas  13517  ressress  13518  rescabs  14025  mgpress  15651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ress 13468
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