Theorem rnun 2644

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

Assertion

Ref	Expression
rnun	⊢ ran (A ∪ B) = (ran A ∪ ran B)

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 2642	. . . 4 ⊢ ◡(A ∪ B) = (◡A ∪ ◡B)
2	1	dmeqi 2532	. . 3 ⊢ dom ◡(A ∪ B) = dom (◡A ∪ ◡B)
3		dmun 2536	. . 3 ⊢ dom (◡A ∪ ◡B) = (dom ◡A ∪ dom ◡B)
4	2, 3	eqtr 1119	. 2 ⊢ dom ◡(A ∪ B) = (dom ◡A ∪ dom ◡B)
5		df-rn 2429	. 2 ⊢ ran (A ∪ B) = dom ◡(A ∪ B)
6		df-rn 2429	. . 3 ⊢ ran A = dom ◡A
7		df-rn 2429	. . 3 ⊢ ran B = dom ◡B
8	6, 7	uneq12i 1609	. 2 ⊢ (ran A ∪ ran B) = (dom ◡A ∪ dom ◡B)
9	4, 5, 8	3eqtr4 1126	1 ⊢ ran (A ∪ B) = (ran A ∪ ran B)

Syntax hints:   = wceq 1091   ∪ cun 1485  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  imaun 2647  fun 2763  sbthlem6 3354  fodomb 3615

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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