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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 26426 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) | |
| 2 | ovex 7464 | . . . . . 6 ⊢ (ℂ ↑m 𝑆) ∈ V | |
| 3 | zex 12622 | . . . . . 6 ⊢ ℤ ∈ V | |
| 4 | 2, 3 | elpm2 8914 | . . . . 5 ⊢ (𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ↔ (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ∧ dom 𝐹 ⊆ ℤ)) |
| 5 | 4 | simplbi 497 | . . . 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 481 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑m 𝑆)) |
| 8 | fndm 6671 | . . . 4 ⊢ (𝐹 Fn 𝑍 → dom 𝐹 = 𝑍) | |
| 9 | 8 | adantr 480 | . . 3 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → dom 𝐹 = 𝑍) |
| 10 | 9 | feq2d 6722 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆))) |
| 11 | 7, 10 | mpbid 232 | 1 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ↑pm cpm 8867 ℂcc 11153 ℤcz 12613 ⇝𝑢culm 26419 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-pm 8869 df-neg 11495 df-z 12614 df-uz 12879 df-ulm 26420 |
| This theorem is referenced by: ulmdvlem1 26443 ulmdvlem2 26444 ulmdvlem3 26445 mtestbdd 26448 mbfulm 26449 iblulm 26450 itgulm 26451 itgulm2 26452 lgamgulm2 27079 lgamcvglem 27083 |
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