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

Theorem rnuni 5091
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  U.  A  =  U_ x  e.  A  ran  x
Distinct variable group:    x, A

Proof of Theorem rnuni
StepHypRef Expression
1 uniiun 3956 . . 3  |-  U. A  =  U_ x  e.  A  x
21rneqi 4904 . 2  |-  ran  U.  A  =  ran  U_  x  e.  A  x
3 rniun 5090 . 2  |-  ran  U_  x  e.  A  x  =  U_ x  e.  A  ran  x
42, 3eqtri 2304 1  |-  ran  U.  A  =  U_ x  e.  A  ran  x
Colors of variables: wff set class
Syntax hints:    = wceq 1624   U.cuni 3828   U_ciun 3906   ran crn 4689
This theorem is referenced by:  ackbij2  7864  axdc3lem2  8072  axfelem21  23767
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-cnv 4696  df-dm 4698  df-rn 4699
  Copyright terms: Public domain W3C validator