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Theorem riinint 5066
Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
riinint  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
Distinct variable groups:    k, V    k, X
Allowed substitution hints:    S( k)    I(
k)

Proof of Theorem riinint
StepHypRef Expression
1 ssexg 4290 . . . . . . 7  |-  ( ( S  C_  X  /\  X  e.  V )  ->  S  e.  _V )
21expcom 425 . . . . . 6  |-  ( X  e.  V  ->  ( S  C_  X  ->  S  e.  _V ) )
32ralimdv 2728 . . . . 5  |-  ( X  e.  V  ->  ( A. k  e.  I  S  C_  X  ->  A. k  e.  I  S  e.  _V ) )
43imp 419 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  A. k  e.  I  S  e.  _V )
5 dfiin3g 5063 . . . 4  |-  ( A. k  e.  I  S  e.  _V  ->  |^|_ k  e.  I  S  =  |^| ran  ( k  e.  I  |->  S ) )
64, 5syl 16 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^|_ k  e.  I  S  =  |^| ran  (
k  e.  I  |->  S ) )
76ineq2d 3485 . 2  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  ( X  i^i  |^| ran  ( k  e.  I  |->  S ) ) )
8 intun 4024 . . 3  |-  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )
9 intsng 4027 . . . . 5  |-  ( X  e.  V  ->  |^| { X }  =  X )
109adantr 452 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| { X }  =  X )
1110ineq1d 3484 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )  =  ( X  i^i  |^| ran  ( k  e.  I  |->  S ) ) )
128, 11syl5eq 2431 . 2  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( X  i^i  |^|
ran  ( k  e.  I  |->  S ) ) )
137, 12eqtr4d 2422 1  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    u. cun 3261    i^i cin 3262    C_ wss 3263   {csn 3757   |^|cint 3992   |^|_ciin 4036    e. cmpt 4207   ran crn 4819
This theorem is referenced by:  cmpfiiin  26442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-int 3993  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-cnv 4826  df-dm 4828  df-rn 4829
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