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Theorem rnun 4947
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 4944 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4740 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmun 4746 . . 3 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
42, 3eqtri 2160 . 2 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
5 df-rn 4550 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4550 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4550 . . 3 ran 𝐵 = dom 𝐵
86, 7uneq12i 3228 . 2 (ran 𝐴 ∪ ran 𝐵) = (dom 𝐴 ∪ dom 𝐵)
94, 5, 83eqtr4i 2170 1 ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  cun 3069  ccnv 4538  dom cdm 4539  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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  imaundi  4951  imaundir  4952  rnpropg  5018  fun  5295  foun  5386  fpr  5602  fprg  5603  sbthlemi6  6850  exmidfodomrlemim  7057
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