Theorem rnun 5699

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 5696	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5480	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5486	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2782	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5277	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5277	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5277	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 3908	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2792	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1632   ∪ cun 3713  ^◡ccnv 5265  dom cdm 5266  ran crn 5267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277

This theorem is referenced by:  imaundi  5703  imaundir  5704  rnpropg  5774  fun  6227  foun  6316  fpr  6584  sbthlem6  8240  fodomr  8276  brwdom2  8643  ordtval  21195  axlowdimlem13  26033  ex-rn  27608  padct  29806  ffsrn  29813  locfinref  30217  esumrnmpt2  30439  noextend  32125  noextendseq  32126  ptrest  33721  rntrclfvOAI  37756  rclexi  38424  rtrclex  38426  rtrclexi  38430  cnvrcl0  38434  rntrcl  38437  dfrtrcl5  38438  dfrcl2  38468  rntrclfv  38526  rnresun  39861