Theorem imaeq12d

Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016)

	Ref	Expression
Hypotheses	imaeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	imaeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	imaeq12d	⊢ ( 𝜑 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		imaeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		imaeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	imaeq1d	⊢ ( 𝜑 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) )
4	2	imaeq2d	⊢ ( 𝜑 → ( 𝐵 “ 𝐶 ) = ( 𝐵 “ 𝐷 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐷 ) )