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Mirrors > Home > MPE Home > Th. List > lmrel | Structured version Visualization version GIF version |
Description: The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
lmrel | ⊢ Rel (⇝𝑡‘𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lm 22576 | . 2 ⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) | |
2 | 1 | relmptopab 7600 | 1 ⊢ Rel (⇝𝑡‘𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 ∀wral 3063 ∃wrex 3072 ∪ cuni 4864 ran crn 5633 ↾ cres 5634 Rel wrel 5637 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 ↑pm cpm 8763 ℂcc 11046 ℤ≥cuz 12760 Topctop 22238 ⇝𝑡clm 22573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fv 6502 df-lm 22576 |
This theorem is referenced by: lmfun 22728 cmetcaulem 24648 lmle 24661 heibor1lem 36257 rrncmslem 36280 xlimrel 44031 |
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