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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 12737 | . . 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 5094 | . 2 ⊢ dom ⇝𝑡 ⊆ Top |
3 | df-br 3977 | . . 3 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) | |
4 | 1 | funmpt2 5221 | . . . . 5 ⊢ Fun ⇝𝑡 |
5 | funrel 5199 | . . . . 5 ⊢ (Fun ⇝𝑡 → Rel ⇝𝑡) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ Rel ⇝𝑡 |
7 | relelfvdm 5512 | . . . 4 ⊢ ((Rel ⇝𝑡 ∧ 〈𝐹, 𝑃〉 ∈ (⇝𝑡‘𝐽)) → 𝐽 ∈ dom ⇝𝑡) | |
8 | 6, 7 | mpan 421 | . . 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 3135 | 1 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 967 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 〈cop 3573 ∪ cuni 3783 class class class wbr 3976 {copab 4036 dom cdm 4598 ran crn 4599 ↾ cres 4600 Rel wrel 4603 Fun wfun 5176 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 ↑pm cpm 6606 ℂcc 7742 ℤ≥cuz 9457 Topctop 12542 ⇝𝑡clm 12734 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fv 5190 df-lm 12737 |
This theorem is referenced by: lmcvg 12764 lmtopcnp 12797 |
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