Theorem lmrcl 23462

Description: Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Assertion

Ref	Expression
lmrcl	⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Proof of Theorem lmrcl

Dummy variables 𝑗 𝑓 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lm 23460	. . 3 ⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
2	1	dmmptss 6241	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 5108	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4		elfvdm 6916	. . 3 ⊢ (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡)
5	3, 4	sylbi 220	. 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡)
6	2, 5	sselid 3932	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  ⟨cop 4593  ∪ cuni 4870   class class class wbr 5107  {copab 5171  dom cdm 5659  ran crn 5660   ↾ cres 5661  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ↑_pm cpm 8831  ℂcc 11126  ℤ_≥cuz 12891  Topctop 23124  ⇝_𝑡clm 23457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fv 6545  df-lm 23460

This theorem is used by:  lmcvg  23493