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Theorem rnuni 5463
 Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rnuni ran 𝐴 = 𝑥𝐴 ran 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem rnuni
StepHypRef Expression
1 uniiun 4509 . . 3 𝐴 = 𝑥𝐴 𝑥
21rneqi 5273 . 2 ran 𝐴 = ran 𝑥𝐴 𝑥
3 rniun 5462 . 2 ran 𝑥𝐴 𝑥 = 𝑥𝐴 ran 𝑥
42, 3eqtri 2632 1 ran 𝐴 = 𝑥𝐴 ran 𝑥
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∪ cuni 4372  ∪ ciun 4455  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  ackbij2  8948  axdc3lem2  9156  unirnmap  38395  unirnmapsn  38401
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