Theorem rnresun 46334

Description: Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
rnresun	⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))

Proof of Theorem rnresun

Step	Hyp	Ref	Expression
1		resundi 5988	. . 3 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
2	1	rneqi 5923	. 2 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
3		rnun 6141	. 2 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
4	2, 3	eqtri 2786	1 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4151  ran crn 5655   ↾ cres 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666

This theorem is used by:  sge0split  47558