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Theorem reluni 5831
Description: The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
reluni (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem reluni
StepHypRef Expression
1 uniiun 5063 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5790 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5829 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 275 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wral 3059   cuni 4912   ciun 4996  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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-ss 3980  df-uni 4913  df-iun 4998  df-rel 5696
This theorem is referenced by:  fununi  6643  frrlem6  8315  wfrrelOLD  8353  tfrlem6  8421  bnj1379  34823
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