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Mirrors > Home > MPE Home > Th. List > lmrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.) |
Ref | Expression |
---|---|
lmrcl | ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lm 21765 | . . 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 6088 | . 2 ⊢ dom ⇝𝑡 ⊆ Top |
3 | df-br 5058 | . . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) | |
4 | elfvdm 6695 | . . 3 ⊢ (〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡) | |
5 | 3, 4 | sylbi 218 | . 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡) |
6 | 2, 5 | sseldi 3962 | 1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 〈cop 4563 ∪ cuni 4830 class class class wbr 5057 {copab 5119 dom cdm 5548 ran crn 5549 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑pm cpm 8396 ℂcc 10523 ℤ≥cuz 12231 Topctop 21429 ⇝𝑡clm 21762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-lm 21765 |
This theorem is referenced by: lmcvg 21798 |
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