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Theorem ressid2 11945
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 967 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  W  e.  X )
32elexd 2673 . . . 4  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  W  e.  _V )
4 simp3 968 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  A  e.  Y )
54elexd 2673 . . . 4  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  A  e.  _V )
6 simp1 966 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  B  C_  A )
76iftrued 3451 . . . . 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 2194 . . . 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 5393 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  ( Base `  W ) )
11 ressbas.b . . . . . . . 8  |-  B  =  ( Base `  W
)
1210, 11syl6eqr 2168 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  ( Base `  w
)  =  B )
13 simpr 109 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  ->  a  =  A )
1412, 13sseq12d 3098 . . . . . 6  |-  ( ( w  =  W  /\  a  =  A )  ->  ( ( Base `  w
)  C_  a  <->  B  C_  A
) )
1513, 12ineq12d 3248 . . . . . . . 8  |-  ( ( w  =  W  /\  a  =  A )  ->  ( a  i^i  ( Base `  w ) )  =  ( A  i^i  B ) )
1615opeq2d 3682 . . . . . . 7  |-  ( ( w  =  W  /\  a  =  A )  -> 
<. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w
) ) >.  =  <. (
Base `  ndx ) ,  ( A  i^i  B
) >. )
179, 16oveq12d 5760 . . . . . 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 3468 . . . . 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 11894 . . . . 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 5868 . . . 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 1201 . . 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 2150 . 2  |-  ( ( B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  W )
231, 22syl5eq 2162 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 947    = wceq 1316    e. wcel 1465   _Vcvv 2660    i^i cin 3040    C_ wss 3041   ifcif 3444   <.cop 3500   ` cfv 5093  (class class class)co 5742   ndxcnx 11883   sSet csts 11884   Basecbs 11886   ↾s cress 11887
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-ress 11894
This theorem is referenced by:  ressid  11947
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