Theorem reldif 5762

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 4089	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		relss 5729	. 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 3902   ⊆ wss 3905  Rel wrel 5628

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-dif 3908  df-ss 3922  df-rel 5630

This theorem is referenced by:  difopab  5777  fundif  6535  relsdom  8886  opeldifid  32561  gsumhashmul  33027  fundmpss  35739  relbigcup  35870  vvdifopab  38234