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Theorem rnuni 5092
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 3955 . . 3  |-  U. A  =  U_ x  e.  A  x
21rneqi 4905 . 2  |-  ran  U. A  =  ran  U_ x  e.  A  x
3 rniun 5091 . 2  |-  ran  U_ x  e.  A  x  =  U_ x  e.  A  ran  x
42, 3eqtri 2303 1  |-  ran  U. A  =  U_ x  e.  A  ran  x
Colors of variables: wff set class
Syntax hints:    = wceq 1623   U.cuni 3827   U_ciun 3905   ran crn 4690
This theorem is referenced by:  ackbij2  7869  axdc3lem2  8077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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