Theorem fcomptss

Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	fcomptss.a	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
	fcomptss.b	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
	fcomptss.g	⊢ ( 𝜑 → 𝐺 : 𝐶 ⟶ 𝐷 )
Assertion	fcomptss	⊢ ( 𝜑 → ( 𝐺 ∘ 𝐹 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		fcomptss.a	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fcomptss.b	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3		fcomptss.g	⊢ ( 𝜑 → 𝐺 : 𝐶 ⟶ 𝐷 )
4	1 2	fssd	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )
5		fcompt	⊢ ( ( 𝐺 : 𝐶 ⟶ 𝐷 ∧ 𝐹 : 𝐴 ⟶ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ) )
6	3 4 5	syl2anc	⊢ ( 𝜑 → ( 𝐺 ∘ 𝐹 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ) )