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

Theorem rnuni 6105
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 5022 . . 3 𝐴 = 𝑥𝐴 𝑥
21rneqi 5896 . 2 ran 𝐴 = ran 𝑥𝐴 𝑥
3 rniun 6104 . 2 ran 𝑥𝐴 𝑥 = 𝑥𝐴 ran 𝑥
42, 3eqtri 2761 1 ran 𝐴 = 𝑥𝐴 ran 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   cuni 4869   ciun 4958  ran crn 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  ackbij2  10187  axdc3lem2  10395  unirnmap  43520  unirnmapsn  43526
  Copyright terms: Public domain W3C validator