Theorem rnun 6103

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 6100	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5853	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5859	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2760	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5635	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5635	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5635	. . 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 5623  dom cdm 5624  ran crn 5625

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 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  imaundi  6107  imaundir  6108  imadifssran  6109  rnpropg  6180  fun  6696  foun  6792  fpr  7101  f1ounsn  7220  sbthlem6  9023  fodomr  9059  fodomfir  9231  brwdom2  9481  ordtval  23164  noextend  27644  noextendseq  27645  axlowdimlem13  29037  ex-rn  30525  padct  32806  ffsrn  32816  esplyind  33734  locfinref  34001  esumrnmpt2  34228  satfrnmapom  35568  ptrest  37954  rntrclfvOAI  43137  tfsconcatrn  43788  rclexi  44060  rtrclex  44062  rtrclexi  44066  cnvrcl0  44070  rntrcl  44073  dfrtrcl5  44074  dfrcl2  44119  rntrclfv  44177  rnresun  45628