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Theorem ulmscl 25548
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 5074 . 2 (𝐹(⇝𝑢𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢𝑆))
2 elfvex 6799 . 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 2106  Vcvv 3429  cop 4567   class class class wbr 5073  cfv 6426  𝑢culm 25545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-dm 5594  df-iota 6384  df-fv 6434
This theorem is referenced by:  ulmcl  25550  ulmf  25551  ulmi  25555  ulmclm  25556  ulmres  25557  ulmshftlem  25558  ulmss  25566  ulmdvlem1  25569  ulmdvlem3  25571  iblulm  25576  itgulm2  25578
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