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

Theorem iinconst 4045
Description: Indexed intersection of a constant class, i.e. where  B does not depend on  x. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
iinconst  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iinconst
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.3rzv 3665 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  A. x  e.  A  y  e.  B )
)
2 vex 2903 . . . 4  |-  y  e. 
_V
3 eliin 4041 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
42, 3ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
51, 4syl6rbbr 256 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  B 
<->  y  e.  B ) )
65eqrdv 2386 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   _Vcvv 2900   (/)c0 3572   |^|_ciin 4037
This theorem is referenced by:  iin0  4315  ptbasfi  17535
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-v 2902  df-dif 3267  df-nul 3573  df-iin 4039
  Copyright terms: Public domain W3C validator