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Theorem ressid2 13404
Description: General behavior of trivial 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
ressid2  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )

Proof of Theorem ressid2
StepHypRef Expression
1 ressbas.r . . . 4  |-  R  =  ( Ws  A )
2 ressbas.b . . . 4  |-  B  =  ( Base `  W
)
31, 2ressval 13403 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iftrue 3660 . . 3  |-  ( B 
C_  A  ->  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  =  W )
53, 4sylan9eqr 2420 . 2  |-  ( ( B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  W )
653impb 1148 1  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    i^i cin 3237    C_ wss 3238   ifcif 3654   <.cop 3732   ` cfv 5358  (class class class)co 5981   ndxcnx 13353   sSet csts 13354   Basecbs 13356   ↾s cress 13357
This theorem is referenced by:  ressbas  13406  resslem  13409  ress0  13410  ressid  13411  ressinbas  13412  ressress  13413  rescabs  13920  mgpress  15546  psgnghm2  26944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-ress 13363
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