ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ressinbasd Unicode version

Theorem ressinbasd 12502
Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressidbasd.1  |-  ( ph  ->  B  =  ( Base `  W ) )
ressidbasd.a  |-  ( ph  ->  A  e.  X )
ressidbasd.w  |-  ( ph  ->  W  e.  V )
Assertion
Ref Expression
ressinbasd  |-  ( ph  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )

Proof of Theorem ressinbasd
StepHypRef Expression
1 ressidbasd.1 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  W ) )
2 inidm 3344 . . . . . . . 8  |-  ( B  i^i  B )  =  B
31ineq2d 3336 . . . . . . . 8  |-  ( ph  ->  ( B  i^i  B
)  =  ( B  i^i  ( Base `  W
) ) )
42, 3eqtr3id 2224 . . . . . . 7  |-  ( ph  ->  B  =  ( B  i^i  ( Base `  W
) ) )
51, 4eqtr3d 2212 . . . . . 6  |-  ( ph  ->  ( Base `  W
)  =  ( B  i^i  ( Base `  W
) ) )
65ineq2d 3336 . . . . 5  |-  ( ph  ->  ( A  i^i  ( Base `  W ) )  =  ( A  i^i  ( B  i^i  ( Base `  W ) ) ) )
7 inass 3345 . . . . 5  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( A  i^i  ( B  i^i  ( Base `  W
) ) )
86, 7eqtr4di 2228 . . . 4  |-  ( ph  ->  ( A  i^i  ( Base `  W ) )  =  ( ( A  i^i  B )  i^i  ( Base `  W
) ) )
98opeq2d 3783 . . 3  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >.  =  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W ) ) >.
)
109oveq2d 5884 . 2  |-  ( ph  ->  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
11 ressidbasd.w . . 3  |-  ( ph  ->  W  e.  V )
12 ressidbasd.a . . 3  |-  ( ph  ->  A  e.  X )
13 ressvalsets 12493 . . 3  |-  ( ( W  e.  V  /\  A  e.  X )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
1411, 12, 13syl2anc 411 . 2  |-  ( ph  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
15 inex1g 4136 . . . 4  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
1612, 15syl 14 . . 3  |-  ( ph  ->  ( A  i^i  B
)  e.  _V )
17 ressvalsets 12493 . . 3  |-  ( ( W  e.  V  /\  ( A  i^i  B )  e.  _V )  -> 
( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
1811, 16, 17syl2anc 411 . 2  |-  ( ph  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
1910, 14, 183eqtr4d 2220 1  |-  ( ph  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128   <.cop 3594   ` cfv 5211  (class class class)co 5868   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: (None)
  Copyright terms: Public domain W3C validator