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Theorem rnuni 5761
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 4763 . . 3 𝐴 = 𝑥𝐴 𝑥
21rneqi 5555 . 2 ran 𝐴 = ran 𝑥𝐴 𝑥
3 rniun 5760 . 2 ran 𝑥𝐴 𝑥 = 𝑥𝐴 ran 𝑥
42, 3eqtri 2821 1 ran 𝐴 = 𝑥𝐴 ran 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653   cuni 4628   ciun 4710  ran crn 5313
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 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  ackbij2  9353  axdc3lem2  9561  unirnmap  40152  unirnmapsn  40158
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