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Theorem rnun 4853
Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
rnun ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)

Proof of Theorem rnun
StepHypRef Expression
1 cnvun 4850 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4650 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmun 4656 . . 3 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
42, 3eqtri 2109 . 2 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
5 df-rn 4463 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4463 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4463 . . 3 ran 𝐵 = dom 𝐵
86, 7uneq12i 3153 . 2 (ran 𝐴 ∪ ran 𝐵) = (dom 𝐴 ∪ dom 𝐵)
94, 5, 83eqtr4i 2119 1 ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1290  cun 2998  ccnv 4451  dom cdm 4452  ran crn 4453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463
This theorem is referenced by:  imaundi  4857  imaundir  4858  rnpropg  4923  fun  5196  foun  5285  fpr  5493  fprg  5494  sbthlemi6  6725  exmidfodomrlemim  6888
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