Theorem lmrcl 14945

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 14943	. . 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 5235	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 4090	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4	1	funmpt2 5367	. . . . 5 ⊢ Fun ⇝𝑡
5		funrel 5345	. . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡)
6	4, 5	ax-mp 5	. . . 4 ⊢ Rel ⇝𝑡
7		relelfvdm 5674	. . . 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 3224	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 1004   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  ⟨cop 3673  ∪ cuni 3894   class class class wbr 4089  {copab 4150  dom cdm 4727  ran crn 4728   ↾ cres 4729  Rel wrel 4732  Fun wfun 5322  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023   ↑_pm cpm 6823  ℂcc 8035  ℤ_≥cuz 9760  Topctop 14750  ⇝_𝑡clm 14940

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fv 5336  df-lm 14943

This theorem is referenced by:  lmcvg  14970  lmtopcnp  15003