Theorem rlimcn1b

Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016)

	Ref	Expression
Hypotheses	rlimcn1b.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑋 )
	rlimcn1b.2	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
	rlimcn1b.3	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
	rlimcn1b.4	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ )
	rlimcn1b.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ 𝑋 ( ( abs ‘ ( 𝑧 − 𝐶 ) ) < 𝑦 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐶 ) ) ) < 𝑥 ) )
Assertion	rlimcn1b	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ⇝𝑟 ( 𝐹 ‘ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rlimcn1b.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑋 )
2		rlimcn1b.2	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
3		rlimcn1b.3	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
4		rlimcn1b.4	⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ )
5		rlimcn1b.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ 𝑋 ( ( abs ‘ ( 𝑧 − 𝐶 ) ) < 𝑦 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐶 ) ) ) < 𝑥 ) )
6	4 1	cofmpt	⊢ ( 𝜑 → ( 𝐹 ∘ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) )
7	1	fmpttd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝑋 )
8	7 2 3 4 5	rlimcn1	⊢ ( 𝜑 → ( 𝐹 ∘ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ⇝𝑟 ( 𝐹 ‘ 𝐶 ) )
9	6 8	eqbrtrrd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝐵 ) ) ⇝𝑟 ( 𝐹 ‘ 𝐶 ) )