Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relin1 Structured version   Visualization version   GIF version

Theorem relin1 5684
 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 4209 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5655 . 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 3939   ⊆ wss 3940  Rel wrel 5559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-in 3947  df-ss 3956  df-rel 5561 This theorem is referenced by:  inopab  5700  idsset  33235  dihmeetlem1N  38293  dihglblem5apreN  38294  dihmeetlem4preN  38309  dihmeetlem13N  38322
 Copyright terms: Public domain W3C validator