Theorem relun 4776

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 3333	. 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
2		df-rel 4666	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
3		df-rel 4666	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	2, 3	anbi12i 460	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5		df-rel 4666	. 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V))
6	1, 4, 5	3bitr4ri 213	1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153   × cxp 4657  Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-rel 4666

This theorem is referenced by:  funun  5298