Theorem rnun 6003

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 6000	. . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
2	1	dmeqi 5772	. . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵)
3		dmun 5778	. . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
4	2, 3	eqtri 2844	. 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
5		df-rn 5565	. 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵)
6		df-rn 5565	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7		df-rn 5565	. . 3 ⊢ ran 𝐵 = dom ◡𝐵
8	6, 7	uneq12i 4136	. 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵)
9	4, 5, 8	3eqtr4i 2854	1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)

Syntax hints:   = wceq 1533   ∪ cun 3933  ^◡ccnv 5553  dom cdm 5554  ran crn 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-cnv 5562  df-dm 5564  df-rn 5565

This theorem is referenced by:  imaundi  6007  imaundir  6008  rnpropg  6078  fun  6539  foun  6632  fpr  6915  sbthlem6  8631  fodomr  8667  brwdom2  9036  ordtval  21796  axlowdimlem13  26739  ex-rn  28218  padct  30454  ffsrn  30464  locfinref  31105  esumrnmpt2  31327  satfrnmapom  32617  noextend  33173  noextendseq  33174  ptrest  34890  rntrclfvOAI  39286  rclexi  39973  rtrclex  39975  rtrclexi  39979  cnvrcl0  39983  rntrcl  39986  dfrtrcl5  39987  dfrcl2  40017  rntrclfv  40075  rnresun  41434