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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 4584 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆)) | |
2 | elfvex 6131 | . 2 ⊢ (〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V) | |
3 | 1, 2 | sylbi 206 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ‘cfv 5804 ⇝𝑢culm 23934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 |
This theorem is referenced by: ulmcl 23939 ulmf 23940 ulmi 23944 ulmclm 23945 ulmres 23946 ulmshftlem 23947 ulmss 23955 ulmdvlem1 23958 ulmdvlem3 23960 iblulm 23965 itgulm2 23967 |
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