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Theorem reldif 5768
Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
reldif (Rel 𝐴 → Rel (𝐴𝐵))

Proof of Theorem reldif
StepHypRef Expression
1 difss 4090 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5734 . 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  cdif 3906  wss 3909  Rel wrel 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-dif 3912  df-in 3916  df-ss 3926  df-rel 5638
This theorem is referenced by:  difopab  5783  difopabOLD  5784  fundif  6546  relsdom  8824  opeldifid  31321  gsumhashmul  31699  fundmpss  34135  relbigcup  34413  vvdifopab  36651
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