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

Theorem reliin 4946
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 4324 . 2  |-  ( E. x  e.  A  B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  B  C_  ( _V  X.  _V ) )
2 df-rel 4832 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
32rexbii 2908 . 2  |-  ( E. x  e.  A  Rel  B  <->  E. x  e.  A  B  C_  ( _V  X.  _V ) )
4 df-rel 4832 . 2  |-  ( Rel  |^|_ x  e.  A  B  <->  |^|_
x  e.  A  B  C_  ( _V  X.  _V ) )
51, 3, 43imtr4i 268 1  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2757   _Vcvv 3061    C_ wss 3416   |^|_ciin 4274    X. cxp 4823   Rel wrel 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-v 3063  df-in 3423  df-ss 3430  df-iin 4276  df-rel 4832
This theorem is referenced by:  relint  4948  xpiindi  4961  dibglbN  34199  dihglbcpreN  34333
  Copyright terms: Public domain W3C validator