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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 5031 | . 2 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆)) | |
2 | elfvex 6678 | . 2 ⊢ (〈𝐹, 𝐺〉 ∈ (⇝𝑢‘𝑆) → 𝑆 ∈ V) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ‘cfv 6324 ⇝𝑢culm 24971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 |
This theorem is referenced by: ulmcl 24976 ulmf 24977 ulmi 24981 ulmclm 24982 ulmres 24983 ulmshftlem 24984 ulmss 24992 ulmdvlem1 24995 ulmdvlem3 24997 iblulm 25002 itgulm2 25004 |
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