Theorem ulmrel 24968

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 24967	. 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 7397	1 ⊢ Rel (⇝𝑢‘𝑆)

Syntax hints:   ∧ w3a 1083  ∀wral 3140  ∃wrex 3141  Vcvv 3496   class class class wbr 5068  Rel wrel 5562  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ↑_m cmap 8408  ℂcc 10537   < clt 10677   − cmin 10872  ℤcz 11984  ℤ_≥cuz 12246  ℝ⁺crp 12392  abscabs 14595  ⇝_𝑢culm 24966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-ulm 24967

This theorem is referenced by:  ulmval  24970  ulmdm  24983  ulmcau  24985  ulmdvlem3  24992