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Theorem relun 5824
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 4200 . 2 ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴𝐵) ⊆ (V × V))
2 df-rel 5696 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5696 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3anbi12i 628 . 2 ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5 df-rel 5696 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
61, 4, 53bitr4ri 304 1 (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  Vcvv 3478  cun 3961  wss 3963   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-rel 5696
This theorem is referenced by:  difxp  6186  funun  6614  fununfun  6616  satfrel  35352
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