Theorem lmrcl 14903

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 14901	. . 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 5229	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 4085	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4	1	funmpt2 5361	. . . . 5 ⊢ Fun ⇝𝑡
5		funrel 5339	. . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡)
6	4, 5	ax-mp 5	. . . 4 ⊢ Rel ⇝𝑡
7		relelfvdm 5665	. . . 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 3223	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1002   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ⟨cop 3670  ∪ cuni 3889   class class class wbr 4084  {copab 4145  dom cdm 4721  ran crn 4722   ↾ cres 4723  Rel wrel 4726  Fun wfun 5316  ⟶wf 5318  ‘cfv 5322  (class class class)co 6011   ↑_pm cpm 6811  ℂcc 8018  ℤ_≥cuz 9743  Topctop 14708  ⇝_𝑡clm 14898

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fv 5330  df-lm 14901

This theorem is referenced by:  lmcvg  14928  lmtopcnp  14961