Theorem hoicoto2

Description: The half-open interval expressed using a composition of a function into ( RR X. RR ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	hoicoto2.i	⊢ ( 𝜑 → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
	hoicoto2.a	⊢ 𝐴 = ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) )
	hoicoto2.b	⊢ 𝐵 = ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) )
Assertion	hoicoto2	⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hoicoto2.i	⊢ ( 𝜑 → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
2		hoicoto2.a	⊢ 𝐴 = ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) )
3		hoicoto2.b	⊢ 𝐵 = ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) )
4	1	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
5		simpr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 )
6	4 5	fvovco	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ) )
7	1	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) )
8		xp1st	⊢ ( ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
9	7 8	syl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
10	9	elexd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ V )
11	2	fvmpt2	⊢ ( ( 𝑘 ∈ 𝑋 ∧ ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ V ) → ( 𝐴 ‘ 𝑘 ) = ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) )
12	5 10 11	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) = ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) )
13	12	eqcomd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) = ( 𝐴 ‘ 𝑘 ) )
14		xp2nd	⊢ ( ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
15	7 14	syl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
16	15	elexd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ V )
17	3	fvmpt2	⊢ ( ( 𝑘 ∈ 𝑋 ∧ ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ V ) → ( 𝐵 ‘ 𝑘 ) = ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) )
18	5 16 17	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) = ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) )
19	18	eqcomd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) = ( 𝐵 ‘ 𝑘 ) )
20	13 19	oveq12d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
21	6 20	eqtrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )
22	21	ixpeq2dva	⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) )