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Mirrors > Home > MPE Home > Th. List > ulmscl | Structured version Visualization version GIF version |
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 5074 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆)) | |
2 | elfvex 6799 | . 2 ⊢ (〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3429 〈cop 4567 class class class wbr 5073 ‘cfv 6426 ⇝𝑢culm 25545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-dm 5594 df-iota 6384 df-fv 6434 |
This theorem is referenced by: ulmcl 25550 ulmf 25551 ulmi 25555 ulmclm 25556 ulmres 25557 ulmshftlem 25558 ulmss 25566 ulmdvlem1 25569 ulmdvlem3 25571 iblulm 25576 itgulm2 25578 |
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