Theorem lmrcl 15044

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 15042	. . 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 5258	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 4109	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4	1	funmpt2 5390	. . . . 5 ⊢ Fun ⇝𝑡
5		funrel 5368	. . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡)
6	4, 5	ax-mp 5	. . . 4 ⊢ Rel ⇝𝑡
7		relelfvdm 5701	. . . 4 ⊢ ((Rel ⇝𝑡 ∧ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽)) → 𝐽 ∈ dom ⇝𝑡)
8	6, 7	mpan 424	. . 3 ⊢ (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡)
9	3, 8	sylbi 121	. 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡)
10	2, 9	sselid 3235	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1005   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  ⟨cop 3691  ∪ cuni 3913   class class class wbr 4108  {copab 4169  dom cdm 4748  ran crn 4749   ↾ cres 4750  Rel wrel 4753  Fun wfun 5345  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049   ↑_pm cpm 6882  ℂcc 8121  ℤ_≥cuz 9849  Topctop 14849  ⇝_𝑡clm 15039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fv 5359  df-lm 15042

This theorem is referenced by:  lmcvg  15069  lmtopcnp  15102