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Theorem lmreltop 14145
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 4771 . 2 Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
2 toptopon2 13971 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 lmfval 14144 . . . 4 (𝐽 ∈ (TopOn‘ 𝐽) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
42, 3sylbi 121 . . 3 (𝐽 ∈ Top → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
54releqd 4728 . 2 (𝐽 ∈ Top → (Rel (⇝𝑡𝐽) ↔ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}))
61, 5mpbiri 168 1 (𝐽 ∈ Top → Rel (⇝𝑡𝐽))
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  cc 7838  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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