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

Theorem reliin 4937
Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
reliin  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )

Proof of Theorem reliin
StepHypRef Expression
1 iinss 4084 . 2  |-  ( E. x  e.  A  B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  B  C_  ( _V  X.  _V ) )
2 df-rel 4826 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
32rexbii 2675 . 2  |-  ( E. x  e.  A  Rel  B  <->  E. x  e.  A  B  C_  ( _V  X.  _V ) )
4 df-rel 4826 . 2  |-  ( Rel  |^|_ x  e.  A  B  <->  |^|_
x  e.  A  B  C_  ( _V  X.  _V ) )
51, 3, 43imtr4i 258 1  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wrex 2651   _Vcvv 2900    C_ wss 3264   |^|_ciin 4037    X. cxp 4817   Rel wrel 4824
This theorem is referenced by:  relint  4939  xpiindi  4951  dibglbN  31282  dihglbcpreN  31416
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-ral 2655  df-rex 2656  df-v 2902  df-in 3271  df-ss 3278  df-iin 4039  df-rel 4826
  Copyright terms: Public domain W3C validator