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Theorem relin1 5474
Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
relin1 (Rel 𝐴 → Rel (𝐴𝐵))

Proof of Theorem relin1
StepHypRef Expression
1 inss1 4059 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5445 . 2 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐴 → Rel (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3797  wss 3798  Rel wrel 5351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805  df-ss 3812  df-rel 5353
This theorem is referenced by:  inopab  5489  idsset  32531  dihmeetlem1N  37360  dihglblem5apreN  37361  dihmeetlem4preN  37376  dihmeetlem13N  37389
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