Theorem lmrel 23259

Description: The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
lmrel	⊢ Rel (⇝𝑡‘𝐽)

Proof of Theorem lmrel

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

Step	Hyp	Ref	Expression
1		df-lm 23258	. 2 ⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
2	1	relmptopab 7700	1 ⊢ Rel (⇝𝑡‘𝐽)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ∪ cuni 4931  ran crn 5701   ↾ cres 5702  Rel wrel 5705  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_pm cpm 8885  ℂcc 11182  ℤ_≥cuz 12903  Topctop 22920  ⇝_𝑡clm 23255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-lm 23258

This theorem is referenced by:  lmfun  23410  cmetcaulem  25341  lmle  25354  heibor1lem  37769  rrncmslem  37792  xlimrel  45741