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Theorem reliin 4823
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 3969 . 2  |-  ( E. x  e.  A  B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  B  C_  ( _V  X.  _V ) )
2 df-rel 4712 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
32rexbii 2581 . 2  |-  ( E. x  e.  A  Rel  B  <->  E. x  e.  A  B  C_  ( _V  X.  _V ) )
4 df-rel 4712 . 2  |-  ( Rel  |^|_ x  e.  A  B  <->  |^|_
x  e.  A  B  C_  ( _V  X.  _V ) )
51, 3, 43imtr4i 257 1  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wrex 2557   _Vcvv 2801    C_ wss 3165   |^|_ciin 3922    X. cxp 4703   Rel wrel 4710
This theorem is referenced by:  relint  4825  xpiindi  4837  dibglbN  31978  dihglbcpreN  32112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-iin 3924  df-rel 4712
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