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Theorem ressid2 14536
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 14535 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iftrue 3945 . . 3  |-  ( B 
C_  A  ->  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  =  W )
53, 4sylan9eqr 2530 . 2  |-  ( ( B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  W )
653impb 1192 1  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   ifcif 3939   <.cop 4033   ` cfv 5586  (class class class)co 6282   ndxcnx 14480   sSet csts 14481   Basecbs 14483   ↾s cress 14484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-ress 14490
This theorem is referenced by:  ressbas  14538  resslem  14541  ress0  14542  ressid  14543  ressinbas  14544  ressress  14545  rescabs  15056  mgpress  16939  psgnghm2  18381  resvsca  27480
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