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 12286 | . . 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 5005 | . 2 ⊢ dom ⇝𝑡 ⊆ Top |
3 | df-br 3900 | . . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) | |
4 | 1 | funmpt2 5132 | . . . . 5 ⊢ Fun ⇝𝑡 |
5 | funrel 5110 | . . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ Rel ⇝𝑡 |
7 | relelfvdm 5421 | . . . 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 3065 | 1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 947 ∈ wcel 1465 ∀wral 2393 ∃wrex 2394 〈cop 3500 ∪ cuni 3706 class class class wbr 3899 {copab 3958 dom cdm 4509 ran crn 4510 ↾ cres 4511 Rel wrel 4514 Fun wfun 5087 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 ↑pm cpm 6511 ℂcc 7586 ℤ≥cuz 9294 Topctop 12091 ⇝𝑡clm 12283 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fv 5101 df-lm 12286 |
This theorem is referenced by: lmcvg 12313 lmtopcnp 12346 |
Copyright terms: Public domain | W3C validator |