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Theorem rnuni 5080
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 3929 . . 3  |-  U. A  =  U_ x  e.  A  x
21rneqi 4893 . 2  |-  ran  U.  A  =  ran  U_  x  e.  A  x
3 rniun 5079 . 2  |-  ran  U_  x  e.  A  x  =  U_ x  e.  A  ran  x
42, 3eqtri 2278 1  |-  ran  U.  A  =  U_ x  e.  A  ran  x
Colors of variables: wff set class
Syntax hints:    = wceq 1619   U.cuni 3801   U_ciun 3879   ran crn 4662
This theorem is referenced by:  ackbij2  7837  axdc3lem2  8045  axfelem21  23735
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-cnv 4677  df-dm 4679  df-rn 4680
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