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Mirrors > Home > MPE Home > Th. List > ulmf2 | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.) |
Ref | Expression |
---|---|
ulmf2 | ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmpm 24970 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) | |
2 | ovex 7188 | . . . . . 6 ⊢ (ℂ ↑m 𝑆) ∈ V | |
3 | zex 11989 | . . . . . 6 ⊢ ℤ ∈ V | |
4 | 2, 3 | elpm2 8437 | . . . . 5 ⊢ (𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ↔ (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ∧ dom 𝐹 ⊆ ℤ)) |
5 | 4 | simplbi 500 | . . . 4 ⊢ (𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) → 𝐹:dom 𝐹⟶(ℂ ↑m 𝑆)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹:dom 𝐹⟶(ℂ ↑m 𝑆)) |
7 | 6 | adantl 484 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑m 𝑆)) |
8 | fndm 6454 | . . . 4 ⊢ (𝐹 Fn 𝑍 → dom 𝐹 = 𝑍) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → dom 𝐹 = 𝑍) |
10 | 9 | feq2d 6499 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆))) |
11 | 7, 10 | mpbid 234 | 1 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5065 dom cdm 5554 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 ↑pm cpm 8406 ℂcc 10534 ℤcz 11980 ⇝𝑢culm 24963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-pm 8408 df-neg 10872 df-z 11981 df-uz 12243 df-ulm 24964 |
This theorem is referenced by: ulmdvlem1 24987 ulmdvlem2 24988 ulmdvlem3 24989 mtestbdd 24992 mbfulm 24993 iblulm 24994 itgulm 24995 itgulm2 24996 lgamgulm2 25612 lgamcvglem 25616 |
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