Theorem rnun 6168

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 6165	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5918	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5924	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2763	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5700	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5700	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5700	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4176	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2773	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1537   ∪ cun 3961  ^◡ccnv 5688  dom cdm 5689  ran crn 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700

This theorem is referenced by:  imaundi  6172  imaundir  6173  rnpropg  6244  fun  6771  foun  6867  fpr  7174  f1ounsn  7292  sbthlem6  9127  fodomr  9167  fodomfir  9366  brwdom2  9611  ordtval  23213  noextend  27726  noextendseq  27727  axlowdimlem13  28984  ex-rn  30469  padct  32737  ffsrn  32747  locfinref  33802  esumrnmpt2  34049  satfrnmapom  35355  ptrest  37606  rntrclfvOAI  42679  tfsconcatrn  43332  rclexi  43605  rtrclex  43607  rtrclexi  43611  cnvrcl0  43615  rntrcl  43618  dfrtrcl5  43619  dfrcl2  43664  rntrclfv  43722  rnresun  45123