Theorem lmrcl 23255

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 23253	. . 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 6263	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 5149	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4		elfvdm 6944	. . 3 ⊢ (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡)
5	3, 4	sylbi 217	. 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡)
6	2, 5	sselid 3993	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ⟨cop 4637  ∪ cuni 4912   class class class wbr 5148  {copab 5210  dom cdm 5689  ran crn 5690   ↾ cres 5691  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_pm cpm 8866  ℂcc 11151  ℤ_≥cuz 12876  Topctop 22915  ⇝_𝑡clm 23250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-lm 23253

This theorem is referenced by:  lmcvg  23286