Theorem ulmscl 26508

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 5114	. 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆))
2		elfvex 6917	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V)
3	1, 2	sylbi 220	1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  ⇝_𝑢culm 26505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-dm 5672  df-iota 6493  df-fv 6545

This theorem is referenced by:  ulmcl  26510  ulmf  26511  ulmi  26515  ulmclm  26516  ulmres  26517  ulmshftlem  26518  ulmss  26526  ulmdvlem1  26529  ulmdvlem3  26531  iblulm  26536  itgulm2  26538