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Theorem resseqnbasd 12501
Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
Hypotheses
Ref Expression
resseqnbas.r  |-  R  =  ( Ws  A )
resseqnbas.e  |-  C  =  ( E `  W
)
resseqnbasd.f  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
resseqnbas.n  |-  ( E `
 ndx )  =/=  ( Base `  ndx )
resseqnbasd.w  |-  ( ph  ->  W  e.  X )
resseqnbasd.a  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
resseqnbasd  |-  ( ph  ->  C  =  ( E `
 R ) )

Proof of Theorem resseqnbasd
StepHypRef Expression
1 resseqnbas.e . 2  |-  C  =  ( E `  W
)
2 resseqnbas.r . . . . 5  |-  R  =  ( Ws  A )
3 resseqnbasd.w . . . . . 6  |-  ( ph  ->  W  e.  X )
4 resseqnbasd.a . . . . . 6  |-  ( ph  ->  A  e.  V )
5 ressvalsets 12493 . . . . . 6  |-  ( ( W  e.  X  /\  A  e.  V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
63, 4, 5syl2anc 411 . . . . 5  |-  ( ph  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
72, 6eqtrid 2222 . . . 4  |-  ( ph  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
87fveq2d 5514 . . 3  |-  ( ph  ->  ( E `  R
)  =  ( E `
 ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) ) )
9 inex1g 4136 . . . . 5  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  W
) )  e.  _V )
104, 9syl 14 . . . 4  |-  ( ph  ->  ( A  i^i  ( Base `  W ) )  e.  _V )
11 resseqnbasd.f . . . . 5  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
12 resseqnbas.n . . . . 5  |-  ( E `
 ndx )  =/=  ( Base `  ndx )
13 basendxnn 12487 . . . . 5  |-  ( Base `  ndx )  e.  NN
1411, 12, 13setsslnid 12483 . . . 4  |-  ( ( W  e.  X  /\  ( A  i^i  ( Base `  W ) )  e.  _V )  -> 
( E `  W
)  =  ( E `
 ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) ) )
153, 10, 14syl2anc 411 . . 3  |-  ( ph  ->  ( E `  W
)  =  ( E `
 ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) ) )
168, 15eqtr4d 2213 . 2  |-  ( ph  ->  ( E `  R
)  =  ( E `
 W ) )
171, 16eqtr4id 2229 1  |-  ( ph  ->  C  =  ( E `
 R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    =/= wne 2347   _Vcvv 2737    i^i cin 3128   <.cop 3594   ` cfv 5211  (class class class)co 5868   NNcn 8895   ndxcnx 12429   sSet csts 12430  Slot cslot 12431   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-nul 3423  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:  ressplusgd  12553
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