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

Theorem reliin 5117
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 4371 . 2  |-  ( E. x  e.  A  B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  B  C_  ( _V  X.  _V ) )
2 df-rel 5001 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
32rexbii 2960 . 2  |-  ( E. x  e.  A  Rel  B  <->  E. x  e.  A  B  C_  ( _V  X.  _V ) )
4 df-rel 5001 . 2  |-  ( Rel  |^|_ x  e.  A  B  <->  |^|_
x  e.  A  B  C_  ( _V  X.  _V ) )
51, 3, 43imtr4i 266 1  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2810   _Vcvv 3108    C_ wss 3471   |^|_ciin 4321    X. cxp 4992   Rel wrel 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-in 3478  df-ss 3485  df-iin 4323  df-rel 5001
This theorem is referenced by:  relint  5119  xpiindi  5131  dibglbN  35840  dihglbcpreN  35974
  Copyright terms: Public domain W3C validator