Theorem fcompt

Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014) (Proof shortened by Mario Carneiro, 27-Dec-2014)

	Ref	Expression
Assertion	fcompt	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝐴 ∘ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ffvelrn	⊢ ( ( 𝐵 : 𝐶 ⟶ 𝐷 ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 )
2	1	adantll	⊢ ( ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 )
3		ffn	⊢ ( 𝐵 : 𝐶 ⟶ 𝐷 → 𝐵 Fn 𝐶 )
4	3	adantl	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 Fn 𝐶 )
5		dffn5	⊢ ( 𝐵 Fn 𝐶 ↔ 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) )
6	4 5	sylib	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) )
7		ffn	⊢ ( 𝐴 : 𝐷 ⟶ 𝐸 → 𝐴 Fn 𝐷 )
8	7	adantr	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 Fn 𝐷 )
9		dffn5	⊢ ( 𝐴 Fn 𝐷 ↔ 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) )
10	8 9	sylib	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) )
11		fveq2	⊢ ( 𝑦 = ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) )
12	2 6 10 11	fmptco	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝐴 ∘ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) )