Theorem rnun 6150

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 6147	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5906	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5912	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2753	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5688	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5688	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5688	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4159	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2763	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1533   ∪ cun 3943  ^◡ccnv 5676  dom cdm 5677  ran crn 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5685  df-dm 5687  df-rn 5688

This theorem is referenced by:  imaundi  6154  imaundir  6155  rnpropg  6226  fun  6757  foun  6854  fpr  7161  sbthlem6  9111  fodomr  9151  brwdom2  9596  ordtval  23123  noextend  27629  noextendseq  27630  axlowdimlem13  28821  ex-rn  30306  padct  32558  ffsrn  32568  locfinref  33512  esumrnmpt2  33757  satfrnmapom  35050  ptrest  37162  rntrclfvOAI  42176  tfsconcatrn  42836  rclexi  43110  rtrclex  43112  rtrclexi  43116  cnvrcl0  43120  rntrcl  43123  dfrtrcl5  43124  dfrcl2  43169  rntrclfv  43227  rnresun  44617