Metamath Proof Explorer


Theorem hoi2toco

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 hoi2toco.1 𝑘 𝜑
hoi2toco.c 𝐼 = ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
Assertion hoi2toco ( 𝜑X 𝑘𝑋 ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = X 𝑘𝑋 ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) ) )

Proof

Step Hyp Ref Expression
1 hoi2toco.1 𝑘 𝜑
2 hoi2toco.c 𝐼 = ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
3 2 funmpt2 Fun 𝐼
4 3 a1i ( 𝜑 → Fun 𝐼 )
5 4 adantr ( ( 𝜑𝑘𝑋 ) → Fun 𝐼 )
6 simpr ( ( 𝜑𝑘𝑋 ) → 𝑘𝑋 )
7 2 dmeqi dom 𝐼 = dom ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
8 7 a1i ( 𝜑 → dom 𝐼 = dom ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) )
9 opex ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V
10 9 2a1i ( 𝜑 → ( 𝑘𝑋 → ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V ) )
11 1 10 ralrimi ( 𝜑 → ∀ 𝑘𝑋 ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V )
12 dmmptg ( ∀ 𝑘𝑋 ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V → dom ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) = 𝑋 )
13 11 12 syl ( 𝜑 → dom ( 𝑘𝑋 ↦ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) = 𝑋 )
14 8 13 eqtr2d ( 𝜑𝑋 = dom 𝐼 )
15 14 adantr ( ( 𝜑𝑘𝑋 ) → 𝑋 = dom 𝐼 )
16 6 15 eleqtrd ( ( 𝜑𝑘𝑋 ) → 𝑘 ∈ dom 𝐼 )
17 fvco ( ( Fun 𝐼𝑘 ∈ dom 𝐼 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( [,) ‘ ( 𝐼𝑘 ) ) )
18 5 16 17 syl2anc ( ( 𝜑𝑘𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( [,) ‘ ( 𝐼𝑘 ) ) )
19 9 a1i ( ( 𝜑𝑘𝑋 ) → ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V )
20 2 fvmpt2 ( ( 𝑘𝑋 ∧ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ∈ V ) → ( 𝐼𝑘 ) = ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
21 6 19 20 syl2anc ( ( 𝜑𝑘𝑋 ) → ( 𝐼𝑘 ) = ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
22 21 fveq2d ( ( 𝜑𝑘𝑋 ) → ( [,) ‘ ( 𝐼𝑘 ) ) = ( [,) ‘ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) )
23 df-ov ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) ) = ( [,) ‘ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ )
24 23 eqcomi ( [,) ‘ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) = ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) )
25 24 a1i ( ( 𝜑𝑘𝑋 ) → ( [,) ‘ ⟨ ( 𝐴𝑘 ) , ( 𝐵𝑘 ) ⟩ ) = ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) ) )
26 18 22 25 3eqtrd ( ( 𝜑𝑘𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) ) )
27 1 26 ixpeq2d ( 𝜑X 𝑘𝑋 ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = X 𝑘𝑋 ( ( 𝐴𝑘 ) [,) ( 𝐵𝑘 ) ) )