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Theorem rnun 5077
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 5074 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4868 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4873 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `'  A  u.  dom  `'  B )
42, 3eqtri 2278 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `'  A  u.  dom  `'  B )
5 df-rn 4680 . 2  |-  ran  (  A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4680 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4680 . . 3  |-  ran  B  =  dom  `'  B
86, 7uneq12i 3302 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `'  A  u.  dom  `'  B )
94, 5, 83eqtr4i 2288 1  |-  ran  (  A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3125   `'ccnv 4660   dom cdm 4661   ran crn 4662
This theorem is referenced by:  imaundi  5081  imaundir  5082  fun  5343  foun  5429  fpr  5638  sbthlem6  6944  fodomr  6980  brwdom2  7255  ordtval  16882  ex-rn  20771  axlowdimlem13  23958
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-cnv 4677  df-dm 4679  df-rn 4680
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