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 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))

Proof of Theorem rnresun

Step	Hyp	Ref	Expression
1		resundi 5748	. . 3 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
2	1	rneqi 5689	. 2 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵))
3		rnun 5880	. 2 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))
4	2, 3	eqtri 2819	1 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵))

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