Theorem imaeq2

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994)

	Ref	Expression
Assertion	imaeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 “ 𝐴 ) = ( 𝐶 “ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		reseq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ↾ 𝐴 ) = ( 𝐶 ↾ 𝐵 ) )
2	1	rneqd	⊢ ( 𝐴 = 𝐵 → ran ( 𝐶 ↾ 𝐴 ) = ran ( 𝐶 ↾ 𝐵 ) )
3		df-ima	⊢ ( 𝐶 “ 𝐴 ) = ran ( 𝐶 ↾ 𝐴 )
4		df-ima	⊢ ( 𝐶 “ 𝐵 ) = ran ( 𝐶 ↾ 𝐵 )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 “ 𝐴 ) = ( 𝐶 “ 𝐵 ) )