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Theorem rnresun 45123
Description: Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
rnresun ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))

Proof of Theorem rnresun
StepHypRef Expression
1 resundi 6014 . . 3 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵))
21rneqi 5951 . 2 ran (𝐹 ↾ (𝐴𝐵)) = ran ((𝐹𝐴) ∪ (𝐹𝐵))
3 rnun 6168 . 2 ran ((𝐹𝐴) ∪ (𝐹𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
42, 3eqtri 2763 1 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961  ran crn 5690  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  sge0split  46365
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