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  23132  noextend  27618  noextendseq  27619  axlowdimlem13  29011  ex-rn  30499  padct  32780  ffsrn  32790  esplyind  33724  locfinref  33991  esumrnmpt2  34218  satfrnmapom  35558  ptrest  37931  rntrclfvOAI  43122  tfsconcatrn  43773  rclexi  44045  rtrclex  44047  rtrclexi  44051  cnvrcl0  44055  rntrcl  44058  dfrtrcl5  44059  dfrcl2  44104  rntrclfv  44162  rnresun  45613