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Theorem relint 4948
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 4946 . 2  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^|_ x  e.  A  x )
2 intiin 4327 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
32releqi 4909 . 2  |-  ( Rel  |^| A  <->  Rel  |^|_ x  e.  A  x )
41, 3sylibr 214 1  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2757   |^|cint 4229   |^|_ciin 4274   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-int 4230  df-iin 4276  df-rel 4832
This theorem is referenced by: (None)
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