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Theorem rnun 5042
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 5039 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4833 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4838 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `'  A  u.  dom  `'  B )
42, 3eqtri 2276 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `'  A  u.  dom  `'  B )
5 df-rn 4645 . 2  |-  ran  (  A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4645 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4645 . . 3  |-  ran  B  =  dom  `'  B
86, 7uneq12i 3269 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `'  A  u.  dom  `'  B )
94, 5, 83eqtr4i 2286 1  |-  ran  (  A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3092   `'ccnv 4625   dom cdm 4626   ran crn 4627
This theorem is referenced by:  imaundi  5046  imaundir  5047  fun  5308  foun  5394  fpr  5603  sbthlem6  6909  fodomr  6945  brwdom2  7220  ordtval  16846  ex-rn  20735  axlowdimlem13  23922
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 2237
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-cnv 4642  df-dm 4644  df-rn 4645
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