Theorem lmreltop 14583

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 4802	. 2 ⊢ Rel {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}
2		toptopon2 14409	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3		lmfval 14582	. . . 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 121	. . 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 4757	. 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 168	1 ⊢ (𝐽 ∈ Top → Rel (⇝𝑡‘𝐽))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484  ∪ cuni 3849  {copab 4103  ran crn 4674   ↾ cres 4675  Rel wrel 4678  ⟶wf 5264  ‘cfv 5268  (class class class)co 5934   ↑_pm cpm 6726  ℂcc 7905  ℤ_≥cuz 9630  Topctop 14387  TopOnctopon 14400  ⇝_𝑡clm 14577

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-pm 6728  df-top 14388  df-topon 14401  df-lm 14580

This theorem is referenced by: (None)