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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 25758 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)) | |
2 | ovex 7395 | . . . . . 6 ⊢ (ℂ ↑m 𝑆) ∈ V | |
3 | zex 12515 | . . . . . 6 ⊢ ℤ ∈ V | |
4 | 2, 3 | elpm2 8819 | . . . . 5 ⊢ (𝐹 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ↔ (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ∧ dom 𝐹 ⊆ ℤ)) |
5 | 4 | simplbi 499 | . . . 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 483 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑m 𝑆)) |
8 | fndm 6610 | . . . 4 ⊢ (𝐹 Fn 𝑍 → dom 𝐹 = 𝑍) | |
9 | 8 | adantr 482 | . . 3 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → dom 𝐹 = 𝑍) |
10 | 9 | feq2d 6659 | . 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → (𝐹:dom 𝐹⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆))) |
11 | 7, 10 | mpbid 231 | 1 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 class class class wbr 5110 dom cdm 5638 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ↑m cmap 8772 ↑pm cpm 8773 ℂcc 11056 ℤcz 12506 ⇝𝑢culm 25751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-pm 8775 df-neg 11395 df-z 12507 df-uz 12771 df-ulm 25752 |
This theorem is referenced by: ulmdvlem1 25775 ulmdvlem2 25776 ulmdvlem3 25777 mtestbdd 25780 mbfulm 25781 iblulm 25782 itgulm 25783 itgulm2 25784 lgamgulm2 26401 lgamcvglem 26405 |
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