Theorem ulmscl 26344

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 5086	. 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆))
2		elfvex 6875	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V)
3	1, 2	sylbi 217	1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  ⇝_𝑢culm 26341

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-dm 5641  df-iota 6454  df-fv 6506

This theorem is referenced by:  ulmcl  26346  ulmf  26347  ulmi  26351  ulmclm  26352  ulmres  26353  ulmshftlem  26354  ulmss  26362  ulmdvlem1  26365  ulmdvlem3  26367  iblulm  26372  itgulm2  26374