Theorem relun 2490

Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.

Assertion

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

Proof of Theorem relun

Step	Hyp	Ref	Expression
1		unss 1632	. 2 ⊢ ((A ⊆ (V × V) ∧ B ⊆ (V × V)) ↔ (A ∪ B) ⊆ (V × V))
2		df-rel 2425	. . 3 ⊢ (Rel A ↔ A ⊆ (V × V))
3		df-rel 2425	. . 3 ⊢ (Rel B ↔ B ⊆ (V × V))
4	2, 3	anbi12i 369	. 2 ⊢ ((Rel A ∧ Rel B) ↔ (A ⊆ (V × V) ∧ B ⊆ (V × V)))
5		df-rel 2425	. 2 ⊢ (Rel (A ∪ B) ↔ (A ∪ B) ⊆ (V × V))
6	1, 4, 5	3bitr4r 159	1 ⊢ (Rel (A ∪ B) ↔ (Rel A ∧ Rel B))

Syntax hints:   ↔ wb 127   ∧ wa 196  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  cnvun 2642  funun 2700

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-rel 2425

metamath.org