ulmf2 - Metamath Proof Explorer

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Theorem ulmf2 24972

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 24971	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
2		ovex 7189	. . . . . 6 ⊢ (ℂ ↑_m 𝑆) ∈ V
3		zex 11991	. . . . . 6 ⊢ ℤ ∈ V
4	2, 3	elpm2 8438	. . . . 5 ⊢ (𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ) ↔ (𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆) ∧ dom 𝐹 ⊆ ℤ))
5	4	simplbi 500	. . . 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 484	. 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → 𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆))
8		fndm 6455	. . . 4 ⊢ (𝐹 Fn 𝑍 → dom 𝐹 = 𝑍)
9	8	adantr 483	. . 3 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → dom 𝐹 = 𝑍)
10	9	feq2d 6500	. 2 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → (𝐹:dom 𝐹⟶(ℂ ↑_m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑_m 𝑆)))
11	7, 10	mpbid 234	1 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝_𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 ↑_pm cpm 8407 ℂcc 10535 ℤcz 11982 ⇝_𝑢culm 24964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-pm 8409 df-neg 10873 df-z 11983 df-uz 12245 df-ulm 24965

This theorem is referenced by: ulmdvlem1 24988 ulmdvlem2 24989 ulmdvlem3 24990 mtestbdd 24993 mbfulm 24994 iblulm 24995 itgulm 24996 itgulm2 24997 lgamgulm2 25613 lgamcvglem 25617

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