Theorem relun 5235

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 3787	. 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
2		df-rel 5121	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3		df-rel 5121	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	2, 3	anbi12i 733	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5		df-rel 5121	. 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
6	1, 4, 5	3bitr4ri 293	1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-rel 5121

This theorem is referenced by:  difxp  5558  funun  5932  fununfun  5934