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Theorem ulmscl 26437
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 5149 . 2 (𝐹(⇝𝑢𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆))
2 elfvex 6945 . 2 (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆) → 𝑆 ∈ V)
31, 2sylbi 217 1 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  cfv 6563  𝑢culm 26434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571
This theorem is referenced by:  ulmcl  26439  ulmf  26440  ulmi  26444  ulmclm  26445  ulmres  26446  ulmshftlem  26447  ulmss  26455  ulmdvlem1  26458  ulmdvlem3  26460  iblulm  26465  itgulm2  26467
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