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Theorem ulmscl 21978
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 4402 . 2  |-  ( F ( ~~> u `  S
) G  <->  <. F ,  G >.  e.  ( ~~> u `  S ) )
2 elfvex 5827 . 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 1758   _Vcvv 3078   <.cop 3992   class class class wbr 4401   ` cfv 5527   ~~> uculm 21975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530  ax-pow 4579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-dm 4959  df-iota 5490  df-fv 5535
This theorem is referenced by:  ulmcl  21980  ulmf  21981  ulmi  21985  ulmclm  21986  ulmres  21987  ulmshftlem  21988  ulmss  21996  ulmdvlem1  21999  ulmdvlem3  22001  iblulm  22006  itgulm2  22008
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