Theorem fcomptf

Description: Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt . (Contributed by Thierry Arnoux, 30-Jun-2017)

	Ref	Expression
Hypothesis	fcomptf.1	⊢ Ⅎ 𝑥 𝐵
Assertion	fcomptf	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝐴 ∘ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		fcomptf.1	⊢ Ⅎ 𝑥 𝐵
2		nfcv	⊢ Ⅎ 𝑥 𝐴
3		nfcv	⊢ Ⅎ 𝑥 𝐷
4		nfcv	⊢ Ⅎ 𝑥 𝐸
5	2 3 4	nff	⊢ Ⅎ 𝑥 𝐴 : 𝐷 ⟶ 𝐸
6		nfcv	⊢ Ⅎ 𝑥 𝐶
7	1 6 3	nff	⊢ Ⅎ 𝑥 𝐵 : 𝐶 ⟶ 𝐷
8	5 7	nfan	⊢ Ⅎ 𝑥 ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 )
9		ffvelcdm	⊢ ( ( 𝐵 : 𝐶 ⟶ 𝐷 ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 )
10	9	adantll	⊢ ( ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 )
11	10	ex	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝑥 ∈ 𝐶 → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 ) )
12	8 11	ralrimi	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ∀ 𝑥 ∈ 𝐶 ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 )
13		ffn	⊢ ( 𝐵 : 𝐶 ⟶ 𝐷 → 𝐵 Fn 𝐶 )
14	13	adantl	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 Fn 𝐶 )
15	1	dffn5f	⊢ ( 𝐵 Fn 𝐶 ↔ 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) )
16	14 15	sylib	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) )
17		ffn	⊢ ( 𝐴 : 𝐷 ⟶ 𝐸 → 𝐴 Fn 𝐷 )
18	17	adantr	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 Fn 𝐷 )
19		dffn5	⊢ ( 𝐴 Fn 𝐷 ↔ 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) )
20	18 19	sylib	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) )
21		fveq2	⊢ ( 𝑦 = ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) )
22	12 16 20 21	fmptcof	⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝐴 ∘ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) )