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Theorem rnun 5089
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 5086 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 4880 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 4885 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `' A  u.  dom  `' B )
42, 3eqtri 2303 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `' A  u.  dom  `' B )
5 df-rn 4700 . 2  |-  ran  ( A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4700 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4700 . . 3  |-  ran  B  =  dom  `' B
86, 7uneq12i 3327 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `' A  u.  dom  `' B )
94, 5, 83eqtr4i 2313 1  |-  ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  imaundi  5093  imaundir  5094  fun  5405  foun  5491  fpr  5704  sbthlem6  6976  fodomr  7012  brwdom2  7287  ordtval  16919  ex-rn  20827  rnpropg  23189  axlowdimlem13  24582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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