Theorem rnun 6104

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 6101	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5854	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5860	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2760	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5636	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5636	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5636	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4119	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2770	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1542   ∪ cun 3900  ^◡ccnv 5624  dom cdm 5625  ran crn 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  imaundi  6108  imaundir  6109  imadifssran  6110  rnpropg  6181  fun  6697  foun  6793  fpr  7101  f1ounsn  7220  sbthlem6  9024  fodomr  9060  fodomfir  9232  brwdom2  9482  ordtval  23137  noextend  27638  noextendseq  27639  axlowdimlem13  29010  ex-rn  30498  padct  32778  ffsrn  32788  esplyind  33712  locfinref  33979  esumrnmpt2  34206  satfrnmapom  35545  ptrest  37791  rntrclfvOAI  42969  tfsconcatrn  43620  rclexi  43892  rtrclex  43894  rtrclexi  43898  cnvrcl0  43902  rntrcl  43905  dfrtrcl5  43906  dfrcl2  43951  rntrclfv  44009  rnresun  45460