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Theorem relun 5268
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 3820 . 2 ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴𝐵) ⊆ (V × V))
2 df-rel 5150 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5150 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3anbi12i 733 . 2 ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5 df-rel 5150 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
61, 4, 53bitr4ri 293 1 (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  Vcvv 3231  cun 3605  wss 3607   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-rel 5150
This theorem is referenced by:  difxp  5593  funun  5970  fununfun  5972
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