Theorem rnun 6118

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 6115	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5868	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5874	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2752	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5649	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5649	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5649	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4129	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2762	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1540   ∪ cun 3912  ^◡ccnv 5637  dom cdm 5638  ran crn 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  imaundi  6122  imaundir  6123  imadifssran  6124  rnpropg  6195  fun  6722  foun  6818  fpr  7126  f1ounsn  7247  sbthlem6  9056  fodomr  9092  fodomfir  9279  brwdom2  9526  ordtval  23076  noextend  27578  noextendseq  27579  axlowdimlem13  28881  ex-rn  30369  padct  32643  ffsrn  32652  locfinref  33831  esumrnmpt2  34058  satfrnmapom  35357  ptrest  37613  rntrclfvOAI  42679  tfsconcatrn  43331  rclexi  43604  rtrclex  43606  rtrclexi  43610  cnvrcl0  43614  rntrcl  43617  dfrtrcl5  43618  dfrcl2  43663  rntrclfv  43721  rnresun  45174