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Theorem reluni 5164
 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 4509 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5125 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5162 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 263 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wral 2896  ∪ cuni 4372  ∪ ciun 4455  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-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-iun 4457  df-rel 5045 This theorem is referenced by:  fununi  5878  wfrrel  7307  tfrlem6  7365  bnj1379  30155  frrlem5b  31029  frrlem6  31033
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