Theorem rnun 4752

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 4749	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4554	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4560	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2101	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4374	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4374	. . 3 |- ran A = dom `' A
7		df-rn 4374	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3124	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2111	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1284   u. cun 2971   `'ccnv 4362   dom cdm 4363   ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  imaundi  4756  imaundir  4757  rnpropg  4820  fun  5083  foun  5165  fpr  5366  fprg  5367