Theorem ulmscl 26321

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 ∈ V)

Proof of Theorem ulmscl

Step	Hyp	Ref	Expression
1		df-br 5103	. 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆))
2		elfvex 6878	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V)
3	1, 2	sylbi 217	1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   class class class wbr 5102  ‘cfv 6499  ⇝_𝑢culm 26318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-dm 5641  df-iota 6452  df-fv 6507

This theorem is referenced by:  ulmcl  26323  ulmf  26324  ulmi  26328  ulmclm  26329  ulmres  26330  ulmshftlem  26331  ulmss  26339  ulmdvlem1  26342  ulmdvlem3  26344  iblulm  26349  itgulm2  26351