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Mirrors > Home > ILE Home > Th. List > lmreltop | GIF version |
Description: The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
lmreltop | ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4624 | . 2 ⊢ Rel {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} | |
2 | toptopon2 12023 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | lmfval 12198 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (⇝𝑡‘𝐽) = {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) | |
4 | 2, 3 | sylbi 120 | . . 3 ⊢ (𝐽 ∈ Top → (⇝𝑡‘𝐽) = {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) |
5 | 4 | releqd 4581 | . 2 ⊢ (𝐽 ∈ Top → (Rel (⇝𝑡‘𝐽) ↔ Rel {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})) |
6 | 1, 5 | mpbiri 167 | 1 ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 943 = wceq 1312 ∈ wcel 1461 ∀wral 2388 ∃wrex 2389 ∪ cuni 3700 {copab 3946 ran crn 4498 ↾ cres 4499 Rel wrel 4502 ⟶wf 5075 ‘cfv 5079 (class class class)co 5726 ↑pm cpm 6495 ℂcc 7539 ℤ≥cuz 9222 Topctop 12001 TopOnctopon 12014 ⇝𝑡clm 12193 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pm 6497 df-top 12002 df-topon 12015 df-lm 12196 |
This theorem is referenced by: (None) |
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