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Theorem rnuni 4950
 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 3866 . . 3 𝐴 = 𝑥𝐴 𝑥
21rneqi 4767 . 2 ran 𝐴 = ran 𝑥𝐴 𝑥
3 rniun 4949 . 2 ran 𝑥𝐴 𝑥 = 𝑥𝐴 ran 𝑥
42, 3eqtri 2160 1 ran 𝐴 = 𝑥𝐴 ran 𝑥
 Colors of variables: wff set class Syntax hints:   = wceq 1331  ∪ cuni 3736  ∪ ciun 3813  ran crn 4540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550 This theorem is referenced by:  ennnfonelemf1  11938
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