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

Theorem rnun 5283
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 5280 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 5074 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 5079 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `' A  u.  dom  `' B )
42, 3eqtri 2458 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `' A  u.  dom  `' B )
5 df-rn 4892 . 2  |-  ran  ( A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4892 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4892 . . 3  |-  ran  B  =  dom  `' B
86, 7uneq12i 3501 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `' A  u.  dom  `' B )
94, 5, 83eqtr4i 2468 1  |-  ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    u. cun 3320   `'ccnv 4880   dom cdm 4881   ran crn 4882
This theorem is referenced by:  imaundi  5287  imaundir  5288  fun  5610  foun  5696  fpr  5917  sbthlem6  7225  fodomr  7261  brwdom2  7544  ordtval  17258  ex-rn  21753  rnpropg  24040  axlowdimlem13  25898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892
  Copyright terms: Public domain W3C validator