Theorem lmbrf

Description: Express the binary relation "sequence F converges to point P " in a metric space using an arbitrary upper set of integers. This version of lmbr2 presupposes that F is a function. (Contributed by Mario Carneiro, 14-Nov-2013)

	Ref	Expression
Hypotheses	lmbr.2	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
	lmbr2.4	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	lmbr2.5	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	lmbrf.6	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑋 )
	lmbrf.7	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐴 )
Assertion	lmbrf	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmbr.2	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
2		lmbr2.4	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3		lmbr2.5	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		lmbrf.6	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑋 )
5		lmbrf.7	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐴 )
6	1 2 3	lmbr2	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) )
7		3anass	⊢ ( ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) )
8	2	uztrn2	⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 )
9	5	eleq1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ↔ 𝐴 ∈ 𝑢 ) )
10	4	fdmd	⊢ ( 𝜑 → dom 𝐹 = 𝑍 )
11	10	eleq2d	⊢ ( 𝜑 → ( 𝑘 ∈ dom 𝐹 ↔ 𝑘 ∈ 𝑍 ) )
12	11	biimpar	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ dom 𝐹 )
13	12	biantrurd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ↔ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
14	9 13	bitr3d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐴 ∈ 𝑢 ↔ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
15	8 14	sylan2	⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐴 ∈ 𝑢 ↔ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
16	15	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐴 ∈ 𝑢 ↔ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
17	16	ralbidva	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
18	17	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
19	18	imbi2d	⊢ ( 𝜑 → ( ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ↔ ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) )
20	19	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) )
21	20	anbi2d	⊢ ( 𝜑 → ( ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) )
22		toponmax	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 ∈ 𝐽 )
23	1 22	syl	⊢ ( 𝜑 → 𝑋 ∈ 𝐽 )
24		cnex	⊢ ℂ ∈ V
25	23 24	jctir	⊢ ( 𝜑 → ( 𝑋 ∈ 𝐽 ∧ ℂ ∈ V ) )
26		uzssz	⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ
27		zsscn	⊢ ℤ ⊆ ℂ
28	26 27	sstri	⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℂ
29	2 28	eqsstri	⊢ 𝑍 ⊆ ℂ
30	4 29	jctir	⊢ ( 𝜑 → ( 𝐹 : 𝑍 ⟶ 𝑋 ∧ 𝑍 ⊆ ℂ ) )
31		elpm2r	⊢ ( ( ( 𝑋 ∈ 𝐽 ∧ ℂ ∈ V ) ∧ ( 𝐹 : 𝑍 ⟶ 𝑋 ∧ 𝑍 ⊆ ℂ ) ) → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) )
32	25 30 31	syl2anc	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) )
33	32	biantrurd	⊢ ( 𝜑 → ( ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) )
34	21 33	bitr2d	⊢ ( 𝜑 → ( ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ) ) )
35	7 34	bitrid	⊢ ( 𝜑 → ( ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ) ) )
36	6 35	bitrd	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝐴 ∈ 𝑢 ) ) ) )