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Theorem ulmscl 24965
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 5050 . 2 (𝐹(⇝𝑢𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆))
2 elfvex 6686 . 2 (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆) → 𝑆 ∈ V)
31, 2sylbi 220 1 (𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3479  cop 4554   class class class wbr 5049  cfv 6338  𝑢culm 24962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5193  ax-pow 5249
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-dm 5548  df-iota 6297  df-fv 6346
This theorem is referenced by:  ulmcl  24967  ulmf  24968  ulmi  24972  ulmclm  24973  ulmres  24974  ulmshftlem  24975  ulmss  24983  ulmdvlem1  24986  ulmdvlem3  24988  iblulm  24993  itgulm2  24995
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