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Theorem relun 5142
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 3743 . 2 ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴𝐵) ⊆ (V × V))
2 df-rel 5030 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5030 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3anbi12i 728 . 2 ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5 df-rel 5030 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
61, 4, 53bitr4ri 291 1 (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  Vcvv 3167  cun 3532  wss 3534   × cxp 5021  Rel wrel 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-un 3539  df-in 3541  df-ss 3548  df-rel 5030
This theorem is referenced by:  difxp  5458  funun  5827  fununfun  5829
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