Theorem rnresun 45420

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 5952	. . 3 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
2	1	rneqi 5886	. 2 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
3		rnun 6103	. 2 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
4	2, 3	eqtri 2759	1 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))

Syntax hints:   = wceq 1541   ∪ cun 3899  ran crn 5625   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636

This theorem is referenced by:  sge0split  46649