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Theorem relint 4912
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 4910 . 2  |-  ( E. x  e.  A  Rel  x  ->  Rel  |^|_ x  e.  A  x )
2 intiin 4058 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
32releqi 4875 . 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 2629   |^|cint 3964   |^|_ciin 4008   Rel wrel 4797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-v 2875  df-in 3245  df-ss 3252  df-int 3965  df-iin 4010  df-rel 4799
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