Theorem reldif 5793

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 4083	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		relss 5758	. 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∖ cdif 3896   ⊆ wss 3899  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-ss 3916  df-rel 5658

This theorem is used by:  difopab  5808  fundif  6581  relsdom  8964  opeldifid  33175  gsumhashmul  33610  fundmpss  36501  relbigcup  36629  vvdifopab  39165