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Theorem lmreltop 12987
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 4738 . 2 Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝐽pm ℂ) ∧ 𝑥 𝐽 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
2 toptopon2 12811 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3 lmfval 12986 . . . 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 4695 . 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 973   = wceq 1348  wcel 2141  wral 2448  wrex 2449   cuni 3796  {copab 4049  ran crn 4612  cres 4613  Rel wrel 4616  wf 5194  cfv 5198  (class class class)co 5853  pm cpm 6627  cc 7772  cuz 9487  Topctop 12789  TopOnctopon 12802  𝑡clm 12981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pm 6629  df-top 12790  df-topon 12803  df-lm 12984
This theorem is referenced by: (None)
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