Theorem reldif 5817

Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Assertion

Ref	Expression
reldif	⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))

Proof of Theorem reldif

Step	Hyp	Ref	Expression
1		difss 4128	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		relss 5783	. 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))

Syntax hints:   → wi 4   ∖ cdif 3941   ⊆ wss 3944  Rel wrel 5683

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3947  df-ss 3961  df-rel 5685

This theorem is referenced by:  difopab  5832  difopabOLD  5833  fundif  6603  relsdom  8971  opeldifid  32468  gsumhashmul  32860  fundmpss  35493  relbigcup  35624  vvdifopab  37862