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

Theorem ulmscl 22940
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 4440 . 2  |-  ( F ( ~~> u `  S
) G  <->  <. F ,  G >.  e.  ( ~~> u `  S ) )
2 elfvex 5875 . 2  |-  ( <. F ,  G >.  e.  ( ~~> u `  S
)  ->  S  e.  _V )
31, 2sylbi 195 1  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   ` cfv 5570   ~~> uculm 22937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568  ax-pow 4615
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-dm 4998  df-iota 5534  df-fv 5578
This theorem is referenced by:  ulmcl  22942  ulmf  22943  ulmi  22947  ulmclm  22948  ulmres  22949  ulmshftlem  22950  ulmss  22958  ulmdvlem1  22961  ulmdvlem3  22963  iblulm  22968  itgulm2  22970
  Copyright terms: Public domain W3C validator