Theorem reldif 5771

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 4076	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		relss 5738	. 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 3886   ⊆ wss 3889  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-ss 3906  df-rel 5638

This theorem is referenced by:  difopab  5786  fundif  6547  relsdom  8900  opeldifid  32669  gsumhashmul  33128  fundmpss  35949  relbigcup  36077  vvdifopab  38586