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Theorem rnresun 40976
 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 5748 . . 3 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵))
21rneqi 5689 . 2 ran (𝐹 ↾ (𝐴𝐵)) = ran ((𝐹𝐴) ∪ (𝐹𝐵))
3 rnun 5880 . 2 ran ((𝐹𝐴) ∪ (𝐹𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
42, 3eqtri 2819 1 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∪ cun 3857  ran crn 5444   ↾ cres 5445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455 This theorem is referenced by:  sge0split  42233
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