Theorem relun 5809

Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
relun	⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Proof of Theorem relun

Step	Hyp	Ref	Expression
1		unss 4183	. 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
2		df-rel 5682	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3		df-rel 5682	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	2, 3	anbi12i 627	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5		df-rel 5682	. 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
6	1, 4, 5	3bitr4ri 303	1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396  Vcvv 3474   ∪ cun 3945   ⊆ wss 3947   × cxp 5673  Rel wrel 5680

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-rel 5682

This theorem is referenced by:  difxp  6160  funun  6591  fununfun  6593  satfrel  34346