Theorem metcnpi

Description: Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp . (Contributed by NM, 17-Dec-2007) (Revised by Mario Carneiro, 13-Nov-2013)

	Ref	Expression
Hypotheses	metcn.2	⊢ 𝐽 = ( MetOpen ‘ 𝐶 )
	metcn.4	⊢ 𝐾 = ( MetOpen ‘ 𝐷 )
Assertion	metcnpi	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝐴 ∈ ℝ+ ) ) → ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		metcn.2	⊢ 𝐽 = ( MetOpen ‘ 𝐶 )
2		metcn.4	⊢ 𝐾 = ( MetOpen ‘ 𝐷 )
3		simpr	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) )
4		simpll	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐶 ∈ ( ∞Met ‘ 𝑋 ) )
5		simplr	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑌 ) )
6		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cnprcl	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝑃 ∈ ∪ 𝐽 )
8	7	adantl	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝑃 ∈ ∪ 𝐽 )
9	1	mopnuni	⊢ ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
10	9	ad2antrr	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝑋 = ∪ 𝐽 )
11	8 10	eleqtrrd	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝑃 ∈ 𝑋 )
12	1 2	metcnp	⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ∧ 𝑃 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑧 ∈ ℝ+ ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) ) ) )
13	4 5 11 12	syl3anc	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑧 ∈ ℝ+ ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) ) ) )
14	3 13	mpbid	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑧 ∈ ℝ+ ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) ) )
15		breq2	⊢ ( 𝑧 = 𝐴 → ( ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ↔ ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) )
16	15	imbi2d	⊢ ( 𝑧 = 𝐴 → ( ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) ↔ ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) ) )
17	16	rexralbidv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) ↔ ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) ) )
18	17	rspccv	⊢ ( ∀ 𝑧 ∈ ℝ+ ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝑧 ) → ( 𝐴 ∈ ℝ+ → ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) ) )
19	14 18	simpl2im	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐴 ∈ ℝ+ → ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) ) )
20	19	impr	⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( ∞Met ‘ 𝑌 ) ) ∧ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝐴 ∈ ℝ+ ) ) → ∃ 𝑥 ∈ ℝ+ ∀ 𝑦 ∈ 𝑋 ( ( 𝑃 𝐶 𝑦 ) < 𝑥 → ( ( 𝐹 ‘ 𝑃 ) 𝐷 ( 𝐹 ‘ 𝑦 ) ) < 𝐴 ) )