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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 4771 | . 2 ⊢ Rel {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))} | |
2 | toptopon2 13971 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | lmfval 14144 | . . . 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 121 | . . 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 4728 | . 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 168 | 1 ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 ∪ cuni 3824 {copab 4078 ran crn 4645 ↾ cres 4646 Rel wrel 4649 ⟶wf 5231 ‘cfv 5235 (class class class)co 5895 ↑pm cpm 6674 ℂcc 7838 ℤ≥cuz 9557 Topctop 13949 TopOnctopon 13962 ⇝𝑡clm 14139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-cnex 7931 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pm 6676 df-top 13950 df-topon 13963 df-lm 14142 |
This theorem is referenced by: (None) |
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