Theorem rnun 6038

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 6035	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5802	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5808	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2766	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5591	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5591	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5591	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4091	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2776	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1539   ∪ cun 3881  ^◡ccnv 5579  dom cdm 5580  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  imaundi  6042  imaundir  6043  rnpropg  6114  fun  6620  foun  6718  fpr  7008  sbthlem6  8828  fodomr  8864  brwdom2  9262  ordtval  22248  axlowdimlem13  27225  ex-rn  28705  padct  30956  ffsrn  30966  locfinref  31693  esumrnmpt2  31936  satfrnmapom  33232  noextend  33796  noextendseq  33797  ptrest  35703  rntrclfvOAI  40429  rclexi  41112  rtrclex  41114  rtrclexi  41118  cnvrcl0  41122  rntrcl  41125  dfrtrcl5  41126  dfrcl2  41171  rntrclfv  41229  rnresun  42605