Theorem ulmscl 26542

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 5110	. 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆))
2		elfvex 6916	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V)
3	1, 2	sylbi 220	1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  ⇝_𝑢culm 26539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-dm 5671  df-iota 6492  df-fv 6544

This theorem is referenced by:  ulmcl  26544  ulmf  26545  ulmi  26549  ulmclm  26550  ulmres  26551  ulmshftlem  26552  ulmss  26560  ulmdvlem1  26563  ulmdvlem3  26565  iblulm  26570  itgulm2  26572