Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > lmrcl | 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 12362 | . . 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 5035 | . 2 ⊢ dom ⇝𝑡 ⊆ Top |
3 | df-br 3930 | . . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) | |
4 | 1 | funmpt2 5162 | . . . . 5 ⊢ Fun ⇝𝑡 |
5 | funrel 5140 | . . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ Rel ⇝𝑡 |
7 | relelfvdm 5453 | . . . 4 ⊢ ((Rel ⇝𝑡 ∧ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) → 𝐽 ∈ dom ⇝𝑡) | |
8 | 6, 7 | mpan 420 | . . 3 ⊢ (〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽) → 𝐽 ∈ dom ⇝𝑡) |
9 | 3, 8 | sylbi 120 | . 2 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ dom ⇝𝑡) |
10 | 2, 9 | sseldi 3095 | 1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 962 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 〈cop 3530 ∪ cuni 3736 class class class wbr 3929 {copab 3988 dom cdm 4539 ran crn 4540 ↾ cres 4541 Rel wrel 4544 Fun wfun 5117 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ↑pm cpm 6543 ℂcc 7621 ℤ≥cuz 9329 Topctop 12167 ⇝𝑡clm 12359 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fv 5131 df-lm 12362 |
This theorem is referenced by: lmcvg 12389 lmtopcnp 12422 |
Copyright terms: Public domain | W3C validator |