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Theorem rnun 5091
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 5088 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4882 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4887 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `' A  u.  dom  `' B )
42, 3eqtri 2305 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `' A  u.  dom  `' B )
5 df-rn 4702 . 2  |-  ran  ( A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4702 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4702 . . 3  |-  ran  B  =  dom  `' B
86, 7uneq12i 3329 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `' A  u.  dom  `' B )
94, 5, 83eqtr4i 2315 1  |-  ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1625    u. cun 3152   `'ccnv 4690   dom cdm 4691   ran crn 4692
This theorem is referenced by:  imaundi  5095  imaundir  5096  fun  5407  foun  5493  fpr  5706  sbthlem6  6978  fodomr  7014  brwdom2  7289  ordtval  16921  ex-rn  20829  rnpropg  23191  axlowdimlem13  24584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-cnv 4699  df-dm 4701  df-rn 4702
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