Theorem mapco2

Description: Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014) (Revised by Stefan O'Rear, 5-May-2015)

	Ref	Expression
Hypothesis	mapco2.3	⊢ 𝐸 ∈ V
Assertion	mapco2	⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m 𝐶 ) ∧ 𝐷 : 𝐸 ⟶ 𝐶 ) → ( 𝐴 ∘ 𝐷 ) ∈ ( 𝐵 ↑m 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		mapco2.3	⊢ 𝐸 ∈ V
2		mapco2g	⊢ ( ( 𝐸 ∈ V ∧ 𝐴 ∈ ( 𝐵 ↑m 𝐶 ) ∧ 𝐷 : 𝐸 ⟶ 𝐶 ) → ( 𝐴 ∘ 𝐷 ) ∈ ( 𝐵 ↑m 𝐸 ) )
3	1 2	mp3an1	⊢ ( ( 𝐴 ∈ ( 𝐵 ↑m 𝐶 ) ∧ 𝐷 : 𝐸 ⟶ 𝐶 ) → ( 𝐴 ∘ 𝐷 ) ∈ ( 𝐵 ↑m 𝐸 ) )