Theorem rnun

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

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

Proof

Step	Hyp	Ref	Expression
1		cnvun	⊢ ◡ ( 𝐴 ∪ 𝐵 ) = ( ◡ 𝐴 ∪ ◡ 𝐵 )
2	1	dmeqi	⊢ dom ◡ ( 𝐴 ∪ 𝐵 ) = dom ( ◡ 𝐴 ∪ ◡ 𝐵 )
3		dmun	⊢ dom ( ◡ 𝐴 ∪ ◡ 𝐵 ) = ( dom ◡ 𝐴 ∪ dom ◡ 𝐵 )
4	2 3	eqtri	⊢ dom ◡ ( 𝐴 ∪ 𝐵 ) = ( dom ◡ 𝐴 ∪ dom ◡ 𝐵 )
5		df-rn	⊢ ran ( 𝐴 ∪ 𝐵 ) = dom ◡ ( 𝐴 ∪ 𝐵 )
6		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
7		df-rn	⊢ ran 𝐵 = dom ◡ 𝐵
8	6 7	uneq12i	⊢ ( ran 𝐴 ∪ ran 𝐵 ) = ( dom ◡ 𝐴 ∪ dom ◡ 𝐵 )
9	4 5 8	3eqtr4i	⊢ ran ( 𝐴 ∪ 𝐵 ) = ( ran 𝐴 ∪ ran 𝐵 )