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Theorem rnun 5460
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 5457 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 5247 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmun 5253 . . 3 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
42, 3eqtri 2632 . 2 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
5 df-rn 5049 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 5049 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 5049 . . 3 ran 𝐵 = dom 𝐵
86, 7uneq12i 3727 . 2 (ran 𝐴 ∪ ran 𝐵) = (dom 𝐴 ∪ dom 𝐵)
94, 5, 83eqtr4i 2642 1 ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cun 3538  ccnv 5037  dom cdm 5038  ran crn 5039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049
This theorem is referenced by:  imaundi  5464  imaundir  5465  rnpropg  5533  fun  5979  foun  6068  fpr  6326  sbthlem6  7960  fodomr  7996  brwdom2  8361  ordtval  20803  axlowdimlem13  25634  ex-rn  26689  padct  28885  ffsrn  28892  locfinref  29236  esumrnmpt2  29457  ptrest  32578  rntrclfvOAI  36272  rclexi  36941  rtrclex  36943  rtrclexi  36947  cnvrcl0  36951  rntrcl  36954  dfrtrcl5  36955  dfrcl2  36985  rntrclfv  37043  rnresun  38357
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