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Theorem ulmscl 26335
Description: Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmscl (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)

Proof of Theorem ulmscl
StepHypRef Expression
1 df-br 5153 . 2 (𝐹(⇝𝑢𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆))
2 elfvex 6940 . 2 (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆) → 𝑆 ∈ V)
31, 2sylbi 216 1 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3473  cop 4638   class class class wbr 5152  cfv 6553  𝑢culm 26332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561
This theorem is referenced by:  ulmcl  26337  ulmf  26338  ulmi  26342  ulmclm  26343  ulmres  26344  ulmshftlem  26345  ulmss  26353  ulmdvlem1  26356  ulmdvlem3  26358  iblulm  26363  itgulm2  26365
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