Theorem rneq

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996)

	Ref	Expression
Assertion	rneq	⊢ ( 𝐴 = 𝐵 → ran 𝐴 = ran 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cnveq	⊢ ( 𝐴 = 𝐵 → ◡ 𝐴 = ◡ 𝐵 )
2	1	dmeqd	⊢ ( 𝐴 = 𝐵 → dom ◡ 𝐴 = dom ◡ 𝐵 )
3		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
4		df-rn	⊢ ran 𝐵 = dom ◡ 𝐵
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ran 𝐴 = ran 𝐵 )