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Theorem relint 4809
Description: The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
relint  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
Distinct variable group:    x, A

Proof of Theorem relint
StepHypRef Expression
1 reliin 4807 . 2  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^|_ x  e.  A  x )
2 intiin 3956 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
32releqi 4772 . 2  |-  ( Rel  |^| A  <->  Rel  |^|_ x  e.  A  x )
41, 3sylibr 203 1  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wrex 2544   |^|cint 3862   |^|_ciin 3906   Rel wrel 4694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-int 3863  df-iin 3908  df-rel 4696
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