Theorem rnun 6144

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 6141	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5901	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5907	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2755	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5683	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5683	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5683	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4157	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2765	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1534   ∪ cun 3942  ^◡ccnv 5671  dom cdm 5672  ran crn 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-cnv 5680  df-dm 5682  df-rn 5683

This theorem is referenced by:  imaundi  6148  imaundir  6149  rnpropg  6220  fun  6753  foun  6851  fpr  7157  sbthlem6  9104  fodomr  9144  brwdom2  9588  ordtval  23080  noextend  27586  noextendseq  27587  axlowdimlem13  28752  ex-rn  30237  padct  32485  ffsrn  32495  locfinref  33378  esumrnmpt2  33623  satfrnmapom  34916  ptrest  37027  rntrclfvOAI  42033  tfsconcatrn  42694  rclexi  42968  rtrclex  42970  rtrclexi  42974  cnvrcl0  42978  rntrcl  42981  dfrtrcl5  42982  dfrcl2  43027  rntrclfv  43085  rnresun  44476