Theorem rnresun 39362

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 u. B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) )

Proof of Theorem rnresun

Step	Hyp	Ref	Expression
1		resundi 5410	. . 3 |- ( F |` ( A u. B ) ) = ( ( F |` A ) u. ( F |` B ) )
2	1	rneqi 5352	. 2 |- ran ( F |` ( A u. B ) ) = ran ( ( F |` A ) u. ( F |` B ) )
3		rnun 5541	. 2 |- ran ( ( F |` A ) u. ( F |` B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) )
4	2, 3	eqtri 2644	1 |- ran ( F |` ( A u. B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) )

Syntax hints:   = wceq 1483   u. cun 3572   ran crn 5115   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  sge0split  40626