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Theorem reluni 5684
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 4973 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5645 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5682 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 277 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wral 3136   cuni 4830   ciun 4910  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-in 3941  df-ss 3950  df-uni 4831  df-iun 4912  df-rel 5555
This theorem is referenced by:  fununi  6422  wfrrel  7952  tfrlem6  8010  bnj1379  32095  frrlem6  33121
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