Theorem lmrcl 12985

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 12984	. . 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 5107	. 2 ⊢ dom ⇝𝑡 ⊆ Top
3		df-br 3990	. . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽))
4	1	funmpt2 5237	. . . . 5 ⊢ Fun ⇝𝑡
5		funrel 5215	. . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡)
6	4, 5	ax-mp 5	. . . 4 ⊢ Rel ⇝𝑡
7		relelfvdm 5528	. . . 4 ⊢ ((Rel ⇝𝑡 ∧ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽)) → 𝐽 ∈ dom ⇝𝑡)
8	6, 7	mpan 422	. . 3 ⊢ (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡)
9	3, 8	sylbi 120	. 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡)
10	2, 9	sselid 3145	1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ w3a 973   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  ⟨cop 3586  ∪ cuni 3796   class class class wbr 3989  {copab 4049  dom cdm 4611  ran crn 4612   ↾ cres 4613  Rel wrel 4616  Fun wfun 5192  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853   ↑_pm cpm 6627  ℂcc 7772  ℤ_≥cuz 9487  Topctop 12789  ⇝_𝑡clm 12981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-lm 12984

This theorem is referenced by:  lmcvg  13011  lmtopcnp  13044