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Theorem relint 4976
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 4974 . 2  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^|_ x  e.  A  x )
2 intiin 4353 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
32releqi 4937 . 2  |-  ( Rel  |^| A  <->  Rel  |^|_ x  e.  A  x )
41, 3sylibr 215 1  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2772   |^|cint 4255   |^|_ciin 4300   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-in 3443  df-ss 3450  df-int 4256  df-iin 4302  df-rel 4860
This theorem is referenced by:  clrellem  36199
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