Theorem imaeq1

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

	Ref	Expression
Assertion	imaeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		reseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ↾ 𝐶 ) = ( 𝐵 ↾ 𝐶 ) )
2	1	rneqd	⊢ ( 𝐴 = 𝐵 → ran ( 𝐴 ↾ 𝐶 ) = ran ( 𝐵 ↾ 𝐶 ) )
3		df-ima	⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
4		df-ima	⊢ ( 𝐵 “ 𝐶 ) = ran ( 𝐵 ↾ 𝐶 )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) )