ulmf2 - Metamath Proof Explorer

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Theorem ulmf2 26338

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 26337	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
2		ovex 7457	. . . . . 6 ⊢ (ℂ ↑_m 𝑆) ∈ V
3		zex 12603	. . . . . 6 ⊢ ℤ ∈ V
4	2, 3	elpm2 8897	. . . . 5 ⊢ (𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ) ↔ (𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆) ∧ dom 𝐹 ⊆ ℤ))
5	4	simplbi 496	. . . 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 480	. 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆))
8		fndm 6660	. . . 4 ⊢ (𝐹 Fn 𝑍 → dom 𝐹 = 𝑍)
9	8	adantr 479	. . 3 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → dom 𝐹 = 𝑍)
10	9	feq2d 6711	. 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 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 class class class wbr 5150 dom cdm 5680 Fn wfn 6546 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ↑_m cmap 8849 ↑_pm cpm 8850 ℂcc 11142 ℤcz 12594 ⇝_𝑢culm 26330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8851 df-pm 8852 df-neg 11483 df-z 12595 df-uz 12859 df-ulm 26331

This theorem is referenced by: ulmdvlem1 26354 ulmdvlem2 26355 ulmdvlem3 26356 mtestbdd 26359 mbfulm 26360 iblulm 26361 itgulm 26362 itgulm2 26363 lgamgulm2 26986 lgamcvglem 26990

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