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Theorem ressvalsets 12493
Description: Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
Assertion
Ref Expression
ressvalsets  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)

Proof of Theorem ressvalsets
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2748 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
21adantr 276 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  W  e.  _V )
3 elex 2748 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
43adantl 277 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  A  e.  _V )
5 simpl 109 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  W  e.  X )
6 basendxnn 12487 . . . 4  |-  ( Base `  ndx )  e.  NN
76a1i 9 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Base `  ndx )  e.  NN )
8 inex1g 4136 . . . 4  |-  ( A  e.  Y  ->  ( A  i^i  ( Base `  W
) )  e.  _V )
98adantl 277 . . 3  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( A  i^i  ( Base `  W ) )  e.  _V )
10 setsex 12464 . . 3  |-  ( ( W  e.  X  /\  ( Base `  ndx )  e.  NN  /\  ( A  i^i  ( Base `  W
) )  e.  _V )  ->  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. )  e.  _V )
115, 7, 9, 10syl3anc 1238 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. )  e.  _V )
12 id 19 . . . 4  |-  ( w  =  W  ->  w  =  W )
13 fveq2 5510 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
1413ineq2d 3336 . . . . 5  |-  ( w  =  W  ->  (
x  i^i  ( Base `  w ) )  =  ( x  i^i  ( Base `  W ) ) )
1514opeq2d 3783 . . . 4  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.  =  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  W ) ) >.
)
1612, 15oveq12d 5886 . . 3  |-  ( w  =  W  ->  (
w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( x  i^i  ( Base `  W ) )
>. ) )
17 ineq1 3329 . . . . 5  |-  ( x  =  A  ->  (
x  i^i  ( Base `  W ) )  =  ( A  i^i  ( Base `  W ) ) )
1817opeq2d 3783 . . . 4  |-  ( x  =  A  ->  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  W ) )
>.  =  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) ) >.
)
1918oveq2d 5884 . . 3  |-  ( x  =  A  ->  ( W sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  W
) ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) )
20 df-iress 12440 . . 3  |-s  =  ( w  e.  _V ,  x  e. 
_V  |->  ( w sSet  <. (
Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
2116, 19, 20ovmpog 6002 . 2  |-  ( ( W  e.  _V  /\  A  e.  _V  /\  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
222, 4, 11, 21syl3anc 1238 1  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128   <.cop 3594   ` cfv 5211  (class class class)co 5868   NNcn 8895   ndxcnx 12429   sSet csts 12430   Basecbs 12432   ↾s cress 12433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-iress 12440
This theorem is referenced by:  ressex  12494  ressval2  12495  ressbasd  12496  strressid  12499  ressval3d  12500  resseqnbasd  12501  ressinbasd  12502  ressressg  12503
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