Theorem rnun 6104

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 6101	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5854	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5860	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2760	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5636	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5636	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5636	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4119	. 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 3900  ^◡ccnv 5624  dom cdm 5625  ran crn 5626

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 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  imaundi  6108  imaundir  6109  imadifssran  6110  rnpropg  6181  fun  6697  foun  6793  fpr  7102  f1ounsn  7221  sbthlem6  9025  fodomr  9061  fodomfir  9233  brwdom2  9483  ordtval  23138  noextend  27639  noextendseq  27640  axlowdimlem13  29032  ex-rn  30520  padct  32800  ffsrn  32810  esplyind  33744  locfinref  34011  esumrnmpt2  34238  satfrnmapom  35577  ptrest  37833  rntrclfvOAI  43011  tfsconcatrn  43662  rclexi  43934  rtrclex  43936  rtrclexi  43940  cnvrcl0  43944  rntrcl  43947  dfrtrcl5  43948  dfrcl2  43993  rntrclfv  44051  rnresun  45502