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Theorem rnuni 4999
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 3853 . . 3  |-  U. A  =  U_ x  e.  A  x
21rneqi 4812 . 2  |-  ran  U.  A  =  ran  U_  x  e.  A  x
3 rniun 4998 . 2  |-  ran  U_  x  e.  A  x  =  U_ x  e.  A  ran  x
42, 3eqtri 2273 1  |-  ran  U.  A  =  U_ x  e.  A  ran  x
Colors of variables: wff set class
Syntax hints:    = wceq 1619   U.cuni 3727   U_ciun 3803   ran crn 4581
This theorem is referenced by:  ackbij2  7753  axdc3lem2  7961  axfelem21  23534
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-cnv 4596  df-dm 4598  df-rn 4599
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