MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnun Unicode version

Theorem rnun 5088
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  u.  B )  =  ( ran  A  u.  ran  B )

Proof of Theorem rnun
StepHypRef Expression
1 cnvun 5085 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4879 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4884 . . 3  |-  dom  ( `' A  u.  `' B )  =  (  dom  `'  A  u.  dom  `'  B )
42, 3eqtri 2304 . 2  |-  dom  `' ( A  u.  B
)  =  (  dom  `'  A  u.  dom  `'  B )
5 df-rn 4699 . 2  |-  ran  (  A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4699 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4699 . . 3  |-  ran  B  =  dom  `'  B
86, 7uneq12i 3328 . 2  |-  ( ran 
A  u.  ran  B
)  =  (  dom  `'  A  u.  dom  `'  B )
94, 5, 83eqtr4i 2314 1  |-  ran  (  A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3151   `'ccnv 4687    dom cdm 4688   ran crn 4689
This theorem is referenced by:  imaundi  5092  imaundir  5093  fun  5371  foun  5457  fpr  5666  sbthlem6  6972  fodomr  7008  brwdom2  7283  ordtval  16915  ex-rn  20804  axlowdimlem13  23992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cnv 4696  df-dm 4698  df-rn 4699
  Copyright terms: Public domain W3C validator