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Theorem relin1 5120
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 3718 . 2  |-  ( A  i^i  B )  C_  A
2 relss 5090 . 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 3475    C_ wss 3476   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-rel 5006
This theorem is referenced by:  inopab  5133  idsset  29145  dihmeetlem1N  36105  dihglblem5apreN  36106  dihmeetlem4preN  36121  dihmeetlem13N  36134
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