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Theorem reluni 5663
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 4945 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5624 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5661 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 278 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wral 3053   cuni 4797   ciun 4882  Rel wrel 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-in 3851  df-ss 3861  df-uni 4798  df-iun 4884  df-rel 5533
This theorem is referenced by:  fununi  6415  wfrrel  7990  tfrlem6  8048  bnj1379  32381  frrlem6  33446
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