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Theorem strressid 12509
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
Hypotheses
Ref Expression
strressid.b  |-  ( ph  ->  B  =  ( Base `  W ) )
strressid.s  |-  ( ph  ->  W Struct  <. M ,  N >. )
strressid.f  |-  ( ph  ->  Fun  W )
strressid.bw  |-  ( ph  ->  ( Base `  ndx )  e.  dom  W )
Assertion
Ref Expression
strressid  |-  ( ph  ->  ( Ws  B )  =  W )

Proof of Theorem strressid
StepHypRef Expression
1 strressid.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  W ) )
21ineq1d 3335 . . . . 5  |-  ( ph  ->  ( B  i^i  ( Base `  W ) )  =  ( ( Base `  W )  i^i  ( Base `  W ) ) )
3 inidm 3344 . . . . 5  |-  ( (
Base `  W )  i^i  ( Base `  W
) )  =  (
Base `  W )
42, 3eqtrdi 2226 . . . 4  |-  ( ph  ->  ( B  i^i  ( Base `  W ) )  =  ( Base `  W
) )
54opeq2d 3783 . . 3  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( B  i^i  ( Base `  W
) ) >.  =  <. (
Base `  ndx ) ,  ( Base `  W
) >. )
65oveq2d 5885 . 2  |-  ( ph  ->  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  W ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( Base `  W ) >. )
)
7 strressid.s . . . 4  |-  ( ph  ->  W Struct  <. M ,  N >. )
8 structex 12454 . . . 4  |-  ( W Struct  <. M ,  N >.  ->  W  e.  _V )
97, 8syl 14 . . 3  |-  ( ph  ->  W  e.  _V )
10 basfn 12499 . . . . 5  |-  Base  Fn  _V
11 funfvex 5528 . . . . . 6  |-  ( ( Fun  Base  /\  W  e. 
dom  Base )  ->  ( Base `  W )  e. 
_V )
1211funfni 5312 . . . . 5  |-  ( (
Base  Fn  _V  /\  W  e.  _V )  ->  ( Base `  W )  e. 
_V )
1310, 9, 12sylancr 414 . . . 4  |-  ( ph  ->  ( Base `  W
)  e.  _V )
141, 13eqeltrd 2254 . . 3  |-  ( ph  ->  B  e.  _V )
15 ressvalsets 12503 . . 3  |-  ( ( W  e.  _V  /\  B  e.  _V )  ->  ( Ws  B )  =  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  W
) ) >. )
)
169, 14, 15syl2anc 411 . 2  |-  ( ph  ->  ( Ws  B )  =  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  W
) ) >. )
)
17 baseid 12495 . . 3  |-  Base  = Slot  ( Base `  ndx )
18 strressid.f . . 3  |-  ( ph  ->  Fun  W )
19 strressid.bw . . 3  |-  ( ph  ->  ( Base `  ndx )  e.  dom  W )
2017, 7, 18, 19strsetsid 12475 . 2  |-  ( ph  ->  W  =  ( W sSet  <. ( Base `  ndx ) ,  ( Base `  W ) >. )
)
216, 16, 203eqtr4d 2220 1  |-  ( ph  ->  ( Ws  B )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128   <.cop 3594   class class class wbr 4000   dom cdm 4623   Fun wfun 5206    Fn wfn 5207   ` cfv 5212  (class class class)co 5869   Struct cstr 12438   ndxcnx 12439   sSet csts 12440   Basecbs 12442   ↾s cress 12443
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 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-distr 7903  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-cnre 7910  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-ltadd 7915
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-sub 8117  df-neg 8118  df-inn 8906  df-n0 9163  df-z 9240  df-uz 9515  df-fz 9993  df-struct 12444  df-ndx 12445  df-slot 12446  df-base 12448  df-sets 12449  df-iress 12450
This theorem is referenced by: (None)
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