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Theorem ulmscl 20285
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 4205 . 2  |-  ( F ( ~~> u `  S
) G  <->  <. F ,  G >.  e.  ( ~~> u `  S ) )
2 elfvex 5750 . 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 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   ` cfv 5446   ~~> uculm 20282
This theorem is referenced by:  ulmcl  20287  ulmf  20288  ulmi  20292  ulmclm  20293  ulmres  20294  ulmshftlem  20295  ulmss  20303  ulmdvlem1  20306  ulmdvlem3  20308  iblulm  20313  itgulm2  20315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454
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