Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ulmscl Structured version   Visualization version   GIF version

Theorem ulmscl 24353
 Description: Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmscl (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)

Proof of Theorem ulmscl
StepHypRef Expression
1 df-br 4787 . 2 (𝐹(⇝𝑢𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆))
2 elfvex 6362 . 2 (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆) → 𝑆 ∈ V)
31, 2sylbi 207 1 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3351  ⟨cop 4322   class class class wbr 4786  ‘cfv 6031  ⇝𝑢culm 24350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-dm 5259  df-iota 5994  df-fv 6039 This theorem is referenced by:  ulmcl  24355  ulmf  24356  ulmi  24360  ulmclm  24361  ulmres  24362  ulmshftlem  24363  ulmss  24371  ulmdvlem1  24374  ulmdvlem3  24376  iblulm  24381  itgulm2  24383
 Copyright terms: Public domain W3C validator