Theorem ulmscl 26595

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 5112	. 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆))
2		elfvex 6920	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V)
3	1, 2	sylbi 220	1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   class class class wbr 5111  ‘cfv 6540  ⇝_𝑢culm 26592

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-dm 5673  df-iota 6496  df-fv 6548

This theorem is used by:  ulmcl  26597  ulmf  26598  ulmi  26602  ulmclm  26603  ulmres  26604  ulmshftlem  26605  ulmss  26613  ulmdvlem1  26616  ulmdvlem3  26618  iblulm  26623  itgulm2  26625