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Theorem ulmscl 20162
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 4154 . 2  |-  ( F ( ~~> u `  S
) G  <->  <. F ,  G >.  e.  ( ~~> u `  S ) )
2 elfvex 5698 . 2  |-  ( <. F ,  G >.  e.  ( ~~> u `  S
)  ->  S  e.  _V )
31, 2sylbi 188 1  |-  ( F ( ~~> u `  S
) G  ->  S  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2899   <.cop 3760   class class class wbr 4153   ` cfv 5394   ~~> uculm 20159
This theorem is referenced by:  ulmcl  20164  ulmf  20165  ulmi  20169  ulmclm  20170  ulmres  20171  ulmshftlem  20172  ulmss  20180  ulmdvlem1  20183  ulmdvlem3  20185  iblulm  20190  itgulm2  20192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279  ax-pow 4318
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-dm 4828  df-iota 5358  df-fv 5402
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