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Theorem rnun 4996
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 4993 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4787 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4792 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `'  A  u.  dom  `'  B )
42, 3eqtri 2273 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `'  A  u.  dom  `'  B )
5 df-rn 4599 . 2  |-  ran  (  A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4599 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4599 . . 3  |-  ran  B  =  dom  `'  B
86, 7uneq12i 3237 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `'  A  u.  dom  `'  B )
94, 5, 83eqtr4i 2283 1  |-  ran  (  A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3076   `'ccnv 4579   dom cdm 4580   ran crn 4581
This theorem is referenced by:  imaundi  5000  imaundir  5001  fun  5262  foun  5348  fpr  5556  sbthlem6  6861  fodomr  6897  brwdom2  7171  ordtval  16751  ex-rn  20640  axlowdimlem13  23756
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-cnv 4596  df-dm 4598  df-rn 4599
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