MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressid2 Unicode version

Theorem ressid2 13476
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 13475 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
4 iftrue 3709 . . 3  |-  ( B 
C_  A  ->  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  =  W )
53, 4sylan9eqr 2462 . 2  |-  ( ( B  C_  A  /\  ( W  e.  X  /\  A  e.  Y
) )  ->  R  =  W )
653impb 1149 1  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   ifcif 3703   <.cop 3781   ` cfv 5417  (class class class)co 6044   ndxcnx 13425   sSet csts 13426   Basecbs 13428   ↾s cress 13429
This theorem is referenced by:  ressbas  13478  resslem  13481  ress0  13482  ressid  13483  ressinbas  13484  ressress  13485  rescabs  13992  mgpress  15618  psgnghm2  27310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-ress 13435
  Copyright terms: Public domain W3C validator