ulmf2 - Metamath Proof Explorer

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Theorem ulmf2 26300

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 26299	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐹 ∈ ((ℂ ↑_m 𝑆) ↑_pm ℤ))
2		ovex 7423	. . . . . 6 ⊢ (ℂ ↑_m 𝑆) ∈ V
3		zex 12545	. . . . . 6 ⊢ ℤ ∈ V
4	2, 3	elpm2 8850	. . . . 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 6624	. . . 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 6675	. 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 1540 ∈ wcel 2109 ⊆ wss 3917 class class class wbr 5110 dom cdm 5641 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 ↑_pm cpm 8803 ℂcc 11073 ℤcz 12536 ⇝_𝑢culm 26292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-map 8804 df-pm 8805 df-neg 11415 df-z 12537 df-uz 12801 df-ulm 26293

This theorem is referenced by: ulmdvlem1 26316 ulmdvlem2 26317 ulmdvlem3 26318 mtestbdd 26321 mbfulm 26322 iblulm 26323 itgulm 26324 itgulm2 26325 lgamgulm2 26953 lgamcvglem 26957

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