o1f - Metamath Proof Explorer

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Theorem o1f 15440

Description: An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)

**Assertion**
Ref	Expression
o1f	⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Proof of Theorem o1f Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elo1 15437	. . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
2	1	simplbi 497	. 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11096	. . . 4 ⊢ ℂ ∈ V
4		reex 11106	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8806	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 497	. 2 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
7	2, 6	syl 17	1 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 ∩ cin 3897 ⊆ wss 3898 class class class wbr 5095 dom cdm 5621 ⟶wf 6484 ‘cfv 6488 (class class class)co 7354 ↑_pm cpm 8759 ℂcc 11013 ℝcr 11014 +∞cpnf 11152 ≤ cle 11156 [,)cico 13251 abscabs 15145 𝑂(1)co1 15397

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-fv 6496 df-ov 7357 df-oprab 7358 df-mpo 7359 df-pm 8761 df-o1 15401

This theorem is referenced by: o1res 15471 o1of2 15524 o1rlimmul 15530 o1mptrcl 15534 o1fsum 15724 o1cxp 26915 dchrisum0 27461

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