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Theorem rnun 5213
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 5210 . . . 4  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
21dmeqi 5004 . . 3  |-  dom  `' ( A  u.  B
)  =  dom  ( `' A  u.  `' B )
3 dmun 5009 . . 3  |-  dom  ( `' A  u.  `' B )  =  ( dom  `' A  u.  dom  `' B )
42, 3eqtri 2400 . 2  |-  dom  `' ( A  u.  B
)  =  ( dom  `' A  u.  dom  `' B )
5 df-rn 4822 . 2  |-  ran  ( A  u.  B )  =  dom  `' ( A  u.  B )
6 df-rn 4822 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4822 . . 3  |-  ran  B  =  dom  `' B
86, 7uneq12i 3435 . 2  |-  ( ran 
A  u.  ran  B
)  =  ( dom  `' A  u.  dom  `' B )
94, 5, 83eqtr4i 2410 1  |-  ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3254   `'ccnv 4810   dom cdm 4811   ran crn 4812
This theorem is referenced by:  imaundi  5217  imaundir  5218  fun  5540  foun  5626  fpr  5846  sbthlem6  7151  fodomr  7187  brwdom2  7467  ordtval  17168  ex-rn  21589  rnpropg  23871  axlowdimlem13  25600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-cnv 4819  df-dm 4821  df-rn 4822
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