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Theorem reluni 5443
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 4761 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5405 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5441 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 267 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wral 3087   cuni 4626   ciun 4708  Rel wrel 5315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-v 3385  df-in 3774  df-ss 3781  df-uni 4627  df-iun 4710  df-rel 5317
This theorem is referenced by:  fununi  6173  wfrrel  7657  tfrlem6  7715  bnj1379  31410  frrlem5b  32290  frrlem6  32294  cnfinltrel  33731
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