Theorem rnun 6135

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 6132	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5885	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5891	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2783	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5661	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5661	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5661	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4165	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2793	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4149  ^◡ccnv 5649  dom cdm 5650  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  imaundi  6140  imaundir  6141  imadifssranOLD  6202  imadifssranOLDOLD  6203  rnpropg  6223  fun  6744  foun  6843  fpr  7158  f1ounsn  7280  sbthlem6  9112  fodomr  9148  fodomfir  9320  brwdom2  9568  ordtval  23508  noextend  28024  noextendseq  28025  axlowdimlem13  29533  ex-rn  31042  padct  33353  ffsrn  33363  esplyind  34250  locfinref  34516  esumrnmpt2  34743  satfrnmapom  36204  ptrest  38613  rntrclfvOAI  43778  tfsconcatrn  44420  rclexi  44691  rtrclex  44693  rtrclexi  44697  cnvrcl0  44701  rntrcl  44704  dfrtrcl5  44705  dfrcl2  44750  rntrclfv  44808  rnresun  46271