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Theorem ressid2 12028
Description: General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
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
Dummy variables  w  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressbas.r . 2  |-  R  =  ( Ws  A )
2 simp2 982 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  W  e.  X )
32elexd 2699 . . . 4  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  W  e.  _V )
4 simp3 983 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  A  e.  Y )
54elexd 2699 . . . 4  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  A  e.  _V )
6 simp1 981 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  B  C_  A )
76iftrued 3481 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  if ( B  C_  A ,  W , 
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )  =  W )
87, 3eqeltrd 2216 . . . 4  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  if ( B  C_  A ,  W , 
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B )
>. ) )  e.  _V )
9 simpl 108 . . . . . . . . 9  |-  ( ( w  =  W  /\  a  =  A )  ->  w  =  W )
109fveq2d 5425 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
11 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
1210, 11syl6eqr 2190 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
13 simpr 109 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
1412, 13sseq12d 3128 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
1513, 12ineq12d 3278 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
1615opeq2d 3712 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
179, 16oveq12d 5792 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
1814, 9, 17ifbieq12d 3498 . . . . 5  |-  ( ( w  =  W  /\  a  =  A )  ->  if ( ( Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >. )
)  =  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
19 df-ress 11977 . . . . 5  |-s  =  ( w  e.  _V ,  a  e. 
_V  |->  if ( (
Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. ) ) )
2018, 19ovmpoga 5900 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V  /\  if ( B  C_  A ,  W ,  ( W sSet  <.
( Base `  ndx ) ,  ( A  i^i  B
) >. ) )  e. 
_V )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
213, 5, 8, 20syl3anc 1216 . . 3  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
) )
2221, 7eqtrd 2172 . 2  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  W )
231, 22syl5eq 2184 1  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070    C_ wss 3071   ifcif 3474   <.cop 3530   ` cfv 5123  (class class class)co 5774   ndxcnx 11966   sSet csts 11967   Basecbs 11969   ↾s cress 11970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-ress 11977
This theorem is referenced by:  ressid  12030
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