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Theorem riinint 4935
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 4160 . . . . . . 7  |-  ( ( S  C_  X  /\  X  e.  V )  ->  S  e.  _V )
21expcom 424 . . . . . 6  |-  ( X  e.  V  ->  ( S  C_  X  ->  S  e.  _V ) )
32ralimdv 2622 . . . . 5  |-  ( X  e.  V  ->  ( A. k  e.  I  S  C_  X  ->  A. k  e.  I  S  e.  _V ) )
43imp 418 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  A. k  e.  I  S  e.  _V )
5 dfiin3g 4932 . . . 4  |-  ( A. k  e.  I  S  e.  _V  ->  |^|_ k  e.  I  S  =  |^| ran  ( k  e.  I  |->  S ) )
64, 5syl 15 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^|_ k  e.  I  S  =  |^| ran  (
k  e.  I  |->  S ) )
76ineq2d 3370 . 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 3894 . . 3  |-  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )
9 intsng 3897 . . . . 5  |-  ( X  e.  V  ->  |^| { X }  =  X )
109adantr 451 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| { X }  =  X )
1110ineq1d 3369 . . 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 2327 . 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 2318 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   {csn 3640   |^|cint 3862   |^|_ciin 3906    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  cmpfiiin  26772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700
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