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Theorem relun 5158
 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 (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Proof of Theorem relun
StepHypRef Expression
1 unss 3749 . 2 ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴𝐵) ⊆ (V × V))
2 df-rel 5045 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5045 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3anbi12i 729 . 2 ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5 df-rel 5045 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
61, 4, 53bitr4ri 292 1 (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540   × cxp 5036  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-rel 5045 This theorem is referenced by:  difxp  5477  funun  5846  fununfun  5848
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