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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 21830 | . . 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 5060 | . . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) | |
4 | elfvdm 6695 | . . 3 ⊢ (〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡) | |
5 | 3, 4 | sylbi 219 | . 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡) |
6 | 2, 5 | sseldi 3958 | 1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 〈cop 4566 ∪ cuni 4831 class class class wbr 5059 {copab 5121 dom cdm 5548 ran crn 5549 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ↑pm cpm 8400 ℂcc 10528 ℤ≥cuz 12237 Topctop 21494 ⇝𝑡clm 21827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 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 21830 |
This theorem is referenced by: lmcvg 21863 |
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