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Theorem reldif 5814
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 4131 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5780 . 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 3945  wss 3948  Rel wrel 5681
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 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3951  df-in 3955  df-ss 3965  df-rel 5683
This theorem is referenced by:  difopab  5829  difopabOLD  5830  fundif  6595  relsdom  8943  opeldifid  31818  gsumhashmul  32196  fundmpss  34727  relbigcup  34858  vvdifopab  37117
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