MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reluni Structured version   Visualization version   GIF version

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 276 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wral 3135   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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-in 3940  df-ss 3949  df-uni 4831  df-iun 4912  df-rel 5555
This theorem is referenced by:  fununi  6422  wfrrel  7949  tfrlem6  8007  bnj1379  32001  frrlem6  33025
  Copyright terms: Public domain W3C validator