ulmf2 - Metamath Proof Explorer

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Theorem ulmf2 26349

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 26348	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
2		ovex 7400	. . . . . 6 ⊢ (ℂ ↑_m 𝑆) ∈ V
3		zex 12533	. . . . . 6 ⊢ ℤ ∈ V
4	2, 3	elpm2 8822	. . . . 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 481	. 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆))
8		fndm 6601	. . . 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 6652	. 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 1542 ∈ wcel 2114 ⊆ wss 3889 class class class wbr 5085 dom cdm 5631 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 ↑_pm cpm 8774 ℂcc 11036 ℤcz 12524 ⇝_𝑢culm 26341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-map 8775 df-pm 8776 df-neg 11380 df-z 12525 df-uz 12789 df-ulm 26342

This theorem is referenced by: ulmdvlem1 26365 ulmdvlem2 26366 ulmdvlem3 26367 mtestbdd 26370 mbfulm 26371 iblulm 26372 itgulm 26373 itgulm2 26374 lgamgulm2 26999 lgamcvglem 27003

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