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Theorem ressval2 13197
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 13195 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iffalse 3572 . . 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 2337 . 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ifcif 3565   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressbas  13198  resslem  13201  ressinbas  13204  ressress  13205  rescabs  13710  mgpress  15336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ress 13155
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