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Theorem reldif 5150
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 3698 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5119 . 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 3536  wss 3539  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-rel 5035
This theorem is referenced by:  difopab  5163  fundif  5835  relsdom  7825  opeldifid  28600  fundmpss  30716  relbigcup  30980
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