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Theorem riinint 4870
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 4126 . . . . . . 7  |-  ( ( S  C_  X  /\  X  e.  V )  ->  S  e.  _V )
21expcom 115 . . . . . 6  |-  ( X  e.  V  ->  ( S  C_  X  ->  S  e.  _V ) )
32ralimdv 2538 . . . . 5  |-  ( X  e.  V  ->  ( A. k  e.  I  S  C_  X  ->  A. k  e.  I  S  e.  _V ) )
43imp 123 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  A. k  e.  I  S  e.  _V )
5 dfiin3g 4867 . . . 4  |-  ( A. k  e.  I  S  e.  _V  ->  |^|_ k  e.  I  S  =  |^| ran  ( k  e.  I  |->  S ) )
64, 5syl 14 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^|_ k  e.  I  S  =  |^| ran  (
k  e.  I  |->  S ) )
76ineq2d 3328 . 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 3860 . . 3  |-  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )
9 intsng 3863 . . . . 5  |-  ( X  e.  V  ->  |^| { X }  =  X )
109adantr 274 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| { X }  =  X )
1110ineq1d 3327 . . 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, 11eqtrid 2215 . 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 2206 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 103    = wceq 1348    e. wcel 2141   A.wral 2448   _Vcvv 2730    u. cun 3119    i^i cin 3120    C_ wss 3121   {csn 3581   |^|cint 3829   |^|_ciin 3872    |-> cmpt 4048   ran crn 4610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-int 3830  df-iin 3874  df-br 3988  df-opab 4049  df-mpt 4050  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by: (None)
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