Theorem lmrel 23270

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 23269	. 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 7642	1 ⊢ Rel (⇝𝑡‘𝐽)

Syntax hints:   → wi 4   ∧ w3a 1097   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ∪ cuni 4864  ran crn 5646   ↾ cres 5647  Rel wrel 5650  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ↑_pm cpm 8804  ℂcc 11068  ℤ_≥cuz 12836  Topctop 22933  ⇝_𝑡clm 23266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525  df-lm 23269

This theorem is referenced by:  lmfun  23421  cmetcaulem  25330  lmle  25343  heibor1lem  38272  rrncmslem  38295  xlimrel  46358