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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 5069 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆)) | |
2 | elfvex 6705 | . 2 ⊢ (〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 〈cop 4575 class class class wbr 5068 ‘cfv 6357 ⇝𝑢culm 24966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 |
This theorem is referenced by: ulmcl 24971 ulmf 24972 ulmi 24976 ulmclm 24977 ulmres 24978 ulmshftlem 24979 ulmss 24987 ulmdvlem1 24990 ulmdvlem3 24992 iblulm 24997 itgulm2 24999 |
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