Theorem constcof

Description: Composition with a constant function. See also fcoconst . (Contributed by Thierry Arnoux, 11-Jan-2026)

	Ref	Expression
Hypotheses	constcof.1	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝐼 )
	constcof.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	constcof	⊢ ( 𝜑 → ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) = ( 𝑋 × { 𝑌 } ) )

Proof

Step	Hyp	Ref	Expression
1		constcof.1	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ 𝐼 )
2		constcof.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
3		fnconstg	⊢ ( 𝑌 ∈ 𝑉 → ( 𝐼 × { 𝑌 } ) Fn 𝐼 )
4	2 3	syl	⊢ ( 𝜑 → ( 𝐼 × { 𝑌 } ) Fn 𝐼 )
5		fnfco	⊢ ( ( ( 𝐼 × { 𝑌 } ) Fn 𝐼 ∧ 𝐹 : 𝑋 ⟶ 𝐼 ) → ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) Fn 𝑋 )
6	4 1 5	syl2anc	⊢ ( 𝜑 → ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) Fn 𝑋 )
7	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐹 : 𝑋 ⟶ 𝐼 )
8		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ 𝑋 )
9	7 8	fvco3d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) ‘ 𝑥 ) = ( ( 𝐼 × { 𝑌 } ) ‘ ( 𝐹 ‘ 𝑥 ) ) )
10	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝑌 ∈ 𝑉 )
11	1	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐼 )
12		fvconst2g	⊢ ( ( 𝑌 ∈ 𝑉 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐼 ) → ( ( 𝐼 × { 𝑌 } ) ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑌 )
13	10 11 12	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝐼 × { 𝑌 } ) ‘ ( 𝐹 ‘ 𝑥 ) ) = 𝑌 )
14	9 13	eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) ‘ 𝑥 ) = 𝑌 )
15	6 14	fconst7v	⊢ ( 𝜑 → ( ( 𝐼 × { 𝑌 } ) ∘ 𝐹 ) = ( 𝑋 × { 𝑌 } ) )