Theorem ulmrel 26436

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 26435	. 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 7683	1 ⊢ Rel (⇝𝑢‘𝑆)

Syntax hints:   ∧ w3a 1086  ∀wral 3059  ∃wrex 3068  Vcvv 3478   class class class wbr 5148  Rel wrel 5694  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  ℂcc 11151   < clt 11293   − cmin 11490  ℤcz 12611  ℤ_≥cuz 12876  ℝ⁺crp 13032  abscabs 15270  ⇝_𝑢culm 26434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ulm 26435

This theorem is referenced by:  ulmval  26438  ulmdm  26451  ulmcau  26453  ulmdvlem3  26460