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Theorem rnuni 4763
 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 3738 . . 3 𝐴 = 𝑥𝐴 𝑥
21rneqi 4590 . 2 ran 𝐴 = ran 𝑥𝐴 𝑥
3 rniun 4762 . 2 ran 𝑥𝐴 𝑥 = 𝑥𝐴 ran 𝑥
42, 3eqtri 2076 1 ran 𝐴 = 𝑥𝐴 ran 𝑥
 Colors of variables: wff set class Syntax hints:   = wceq 1259  ∪ cuni 3608  ∪ ciun 3685  ran crn 4374 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384 This theorem is referenced by: (None)
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