Theorem rnun 3457

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60.

Assertion

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

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 3455	. . . 4 |- `'(A u. B) = (`'A u. `'B)
2	1	dmeqi 3312	. . 3 |- dom `'(A u. B) = dom (`'A u. `'B)
3		dmun 3317	. . 3 |- dom (`'A u. `'B) = (dom `' A u. dom `' B)
4	2, 3	eqtr 1495	. 2 |- dom `'(A u. B) = (dom `' A u. dom `' B)
5		df-rn 3189	. 2 |- ran ( A u. B) = dom `'(A u. B)
6		df-rn 3189	. . 3 |- ran A = dom `' A
7		df-rn 3189	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 2182	. 2 |- (ran A u. ran B) = (dom `' A u. dom `' B)
9	4, 5, 8	3eqtr4 1505	1 |- ran ( A u. B) = (ran A u. ran B)

Syntax hints:   = wceq 956   u. cun 2045  `'ccnv 3169  dom cdm 3170  ran crn 3171

This theorem is referenced by:  imaun 3460  imaun2 3461  fun 3641  sbthlem6 4452  fodomr 4483

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189

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