Theorem reldif 5737

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 4072	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		relss 5703	. 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 3889   ⊆ wss 3892  Rel wrel 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-dif 3895  df-in 3899  df-ss 3909  df-rel 5607

This theorem is referenced by:  difopab  5752  difopabOLD  5753  fundif  6512  relsdom  8771  opeldifid  30987  gsumhashmul  31365  fundmpss  33789  relbigcup  34248  vvdifopab  36470