Theorem rnun 6101

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 6098	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5851	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5857	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2760	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5633	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5633	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5633	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4107	. 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 3888  ^◡ccnv 5621  dom cdm 5622  ran crn 5623

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  imaundi  6105  imaundir  6106  imadifssran  6107  rnpropg  6178  fun  6694  foun  6790  fpr  7099  f1ounsn  7218  sbthlem6  9021  fodomr  9057  fodomfir  9229  brwdom2  9479  ordtval  23163  noextend  27649  noextendseq  27650  axlowdimlem13  29042  ex-rn  30530  padct  32811  ffsrn  32821  esplyind  33739  locfinref  34006  esumrnmpt2  34233  satfrnmapom  35573  ptrest  37951  rntrclfvOAI  43134  tfsconcatrn  43785  rclexi  44057  rtrclex  44059  rtrclexi  44063  cnvrcl0  44067  rntrcl  44070  dfrtrcl5  44071  dfrcl2  44116  rntrclfv  44174  rnresun  45625