MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riinint Unicode version

Theorem riinint 4951
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 4176 . . . . . . 7  |-  ( ( S  C_  X  /\  X  e.  V )  ->  S  e.  _V )
21expcom 424 . . . . . 6  |-  ( X  e.  V  ->  ( S  C_  X  ->  S  e.  _V ) )
32ralimdv 2635 . . . . 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 4948 . . . 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 3383 . 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 3910 . . 3  |-  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )
9 intsng 3913 . . . . 5  |-  ( X  e.  V  ->  |^| { X }  =  X )
109adantr 451 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| { X }  =  X )
1110ineq1d 3382 . . 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 2340 . 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 2331 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   {csn 3653   |^|cint 3878   |^|_ciin 3922    e. cmpt 4093   ran crn 4706
This theorem is referenced by:  cmpfiiin  26875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-int 3879  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716
  Copyright terms: Public domain W3C validator