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| Description: Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| ulmscl | ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5143 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆)) | |
| 2 | elfvex 6943 | . 2 ⊢ (〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 〈cop 4631 class class class wbr 5142 ‘cfv 6560 ⇝𝑢culm 26420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: ulmcl 26425 ulmf 26426 ulmi 26430 ulmclm 26431 ulmres 26432 ulmshftlem 26433 ulmss 26441 ulmdvlem1 26444 ulmdvlem3 26446 iblulm 26451 itgulm2 26453 | 
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