MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riin0 Unicode version

Theorem riin0 3975
Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riin0  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Distinct variable groups:    x, A    x, X
Allowed substitution hint:    S( x)

Proof of Theorem riin0
StepHypRef Expression
1 iineq1 3919 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3370 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 3960 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3367 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3481 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2303 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6syl6eq 2331 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   _Vcvv 2788    i^i cin 3151   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  riinrab  3977  riiner  6732  mreriincl  13500  riinopn  16654  riincld  16781  fnemeet2  26316  pmapglb2N  29960  pmapglb2xN  29961
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iin 3908
  Copyright terms: Public domain W3C validator