Theorem ulmrel 26399

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 26398	. 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 7675	1 ⊢ Rel (⇝𝑢‘𝑆)

Syntax hints:   ∧ w3a 1084  ∀wral 3050  ∃wrex 3059  Vcvv 3461   class class class wbr 5152  Rel wrel 5686  ⟶wf 6549  ‘cfv 6553  (class class class)co 7423   ↑_m cmap 8854  ℂcc 11152   < clt 11294   − cmin 11490  ℤcz 12605  ℤ_≥cuz 12869  ℝ⁺crp 13023  abscabs 15234  ⇝_𝑢culm 26397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ulm 26398

This theorem is referenced by:  ulmval  26401  ulmdm  26414  ulmcau  26416  ulmdvlem3  26423