ulmf2 - Metamath Proof Explorer

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Theorem ulmf2 26441

Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)

**Assertion**
Ref	Expression
ulmf2	⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))

**Proof of Theorem ulmf2**
Step	Hyp	Ref	Expression
1		ulmpm 26440	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
2		ovex 7463	. . . . . 6 ⊢ (ℂ ↑_m 𝑆) ∈ V
3		zex 12619	. . . . . 6 ⊢ ℤ ∈ V
4	2, 3	elpm2 8912	. . . . 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 1536 ∈ wcel 2105 ⊆ wss 3962 class class class wbr 5147 dom cdm 5688 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑_m cmap 8864 ↑_pm cpm 8865 ℂcc 11150 ℤcz 12610 ⇝_𝑢culm 26433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-pm 8867 df-neg 11492 df-z 12611 df-uz 12876 df-ulm 26434

This theorem is referenced by: ulmdvlem1 26457 ulmdvlem2 26458 ulmdvlem3 26459 mtestbdd 26462 mbfulm 26463 iblulm 26464 itgulm 26465 itgulm2 26466 lgamgulm2 27093 lgamcvglem 27097

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