Theorem ulmrel 26422

Description: The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)

Assertion

Ref	Expression
ulmrel	⊢ Rel (⇝𝑢‘𝑆)

Proof of Theorem ulmrel

Dummy variables 𝑓 𝑗 𝑘 𝑛 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ulm 26421	. 2 ⊢ ⇝𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)})
2	1	relmptopab 7684	1 ⊢ Rel (⇝𝑢‘𝑆)

Syntax hints:   ∧ w3a 1086  ∀wral 3060  ∃wrex 3069  Vcvv 3479   class class class wbr 5142  Rel wrel 5689  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  ℂcc 11154   < clt 11296   − cmin 11493  ℤcz 12615  ℤ_≥cuz 12879  ℝ⁺crp 13035  abscabs 15274  ⇝_𝑢culm 26420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ulm 26421

This theorem is referenced by:  ulmval  26424  ulmdm  26437  ulmcau  26439  ulmdvlem3  26446