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 (⇝𝑡‘𝐽))

Proof of Theorem lmreltop

Dummy variables 𝑓 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4624	. 2 ⊢ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}
2		toptopon2 12023	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3		lmfval 12198	. . . 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 120	. . 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 4581	. 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 167	1 ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽))

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  ℂcc 7539  ℤ_≥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)