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Theorem lmreltop 12199
Description: The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
Assertion
Ref Expression
lmreltop (𝐽 ∈ Top → Rel (⇝𝑡𝐽))

Proof of Theorem lmreltop
Dummy variables 𝑓 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4624 . 2 Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
2 toptopon2 12023 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 lmfval 12198 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
42, 3sylbi 120 . . 3 (𝐽 ∈ Top → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
54releqd 4581 . 2 (𝐽 ∈ Top → (Rel (⇝𝑡𝐽) ↔ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}))
61, 5mpbiri 167 1 (𝐽 ∈ Top → Rel (⇝𝑡𝐽))
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  cc 7539  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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