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Theorem relint 4962
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 4960 . 2  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^|_ x  e.  A  x )
2 intiin 4323 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
32releqi 4923 . 2  |-  ( Rel  |^| A  <->  Rel  |^|_ x  e.  A  x )
41, 3sylibr 217 1  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2757   |^|cint 4226   |^|_ciin 4270   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-int 4227  df-iin 4272  df-rel 4846
This theorem is referenced by:  clrellem  36300
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