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

Theorem ulmscl 19758
Description: Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmscl  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )

Proof of Theorem ulmscl
StepHypRef Expression
1 df-br 4024 . 2  |-  ( F ( ~~> u `  S
) G  <->  <. F ,  G >.  e.  ( ~~> u `  S ) )
2 elfvex 5555 . 2  |-  ( <. F ,  G >.  e.  ( ~~> u `  S
)  ->  S  e.  _V )
31, 2sylbi 187 1  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   ` cfv 5255   ~~> uculm 19755
This theorem is referenced by:  ulmcl  19760  ulmf  19761  ulmi  19765  ulmclm  19766  ulmres  19767  ulmshftlem  19768  ulmss  19774  ulmdvlem1  19777  ulmdvlem3  19779  iblulm  19783  itgulm2  19785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263
  Copyright terms: Public domain W3C validator