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

Theorem reluni 5380
 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 4705 . . 3 𝐴 = 𝑥𝐴 𝑥
21releqi 5342 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
3 reliun 5378 . 2 (Rel 𝑥𝐴 𝑥 ↔ ∀𝑥𝐴 Rel 𝑥)
42, 3bitri 264 1 (Rel 𝐴 ↔ ∀𝑥𝐴 Rel 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wral 3060  ∪ cuni 4572  ∪ ciun 4652  Rel wrel 5254 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-uni 4573  df-iun 4654  df-rel 5256 This theorem is referenced by:  fununi  6104  wfrrel  7572  tfrlem6  7630  bnj1379  31233  frrlem5b  32116  frrlem6  32120  cnfinltrel  33571
 Copyright terms: Public domain W3C validator