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Theorem relin1 5110
Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
relin1  |-  ( Rel 
A  ->  Rel  ( A  i^i  B ) )

Proof of Theorem relin1
StepHypRef Expression
1 inss1 3703 . 2  |-  ( A  i^i  B )  C_  A
2 relss 5080 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( Rel  A  ->  Rel  ( A  i^i  B ) ) )
31, 2ax-mp 5 1  |-  ( Rel 
A  ->  Rel  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    i^i cin 3460    C_ wss 3461   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-in 3468  df-ss 3475  df-rel 4996
This theorem is referenced by:  inopab  5123  idsset  29515  dihmeetlem1N  36757  dihglblem5apreN  36758  dihmeetlem4preN  36773  dihmeetlem13N  36786
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