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Theorem rnun 5171
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 5168 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4962 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4967 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `' A  u.  dom  `' B )
42, 3eqtri 2378 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `' A  u.  dom  `' B )
5 df-rn 4782 . 2  |-  ran  ( A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4782 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4782 . . 3  |-  ran  B  =  dom  `' B
86, 7uneq12i 3403 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `' A  u.  dom  `' B )
94, 5, 83eqtr4i 2388 1  |-  ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1642    u. cun 3226   `'ccnv 4770   dom cdm 4771   ran crn 4772
This theorem is referenced by:  imaundi  5175  imaundir  5176  fun  5488  foun  5574  fpr  5787  sbthlem6  7064  fodomr  7100  brwdom2  7377  ordtval  17025  ex-rn  20939  rnpropg  23233  axlowdimlem13  25141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-cnv 4779  df-dm 4781  df-rn 4782
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