Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnresun Structured version   Visualization version   GIF version

Theorem rnresun 39879
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 5568 . . 3 (𝐹 ↾ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵))
21rneqi 5507 . 2 ran (𝐹 ↾ (𝐴𝐵)) = ran ((𝐹𝐴) ∪ (𝐹𝐵))
3 rnun 5699 . 2 ran ((𝐹𝐴) ∪ (𝐹𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
42, 3eqtri 2782 1 ran (𝐹 ↾ (𝐴𝐵)) = (ran (𝐹𝐴) ∪ ran (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cun 3713  ran crn 5267  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278
This theorem is referenced by:  sge0split  41147
  Copyright terms: Public domain W3C validator