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Theorem ulmrel 24982
Description: The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmrel Rel (⇝𝑢𝑆)

Proof of Theorem ulmrel
Dummy variables 𝑓 𝑗 𝑘 𝑛 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ulm 24981 . 2 𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
21relmptopab 7391 1 Rel (⇝𝑢𝑆)
Colors of variables: wff setvar class
Syntax hints:  w3a 1084  wral 3133  wrex 3134  Vcvv 3480   class class class wbr 5053  Rel wrel 5548  wf 6341  cfv 6345  (class class class)co 7151  m cmap 8404  cc 10535   < clt 10675  cmin 10870  cz 11980  cuz 12242  +crp 12388  abscabs 14595  𝑢culm 24980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fv 6353  df-ulm 24981
This theorem is referenced by:  ulmval  24984  ulmdm  24997  ulmcau  24999  ulmdvlem3  25006
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