o1f - Metamath Proof Explorer

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Theorem o1f 15486

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 15483	. . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
2	1	simplbi 498	. 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11115	. . . 4 ⊢ ℂ ∈ V
4		reex 11125	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8816	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 498	. 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 2121 ∀wral 3055 ∃wrex 3065 ∩ cin 3883 ⊆ wss 3884 class class class wbr 5074 dom cdm 5620 ⟶wf 6484 ‘cfv 6488 (class class class)co 7359 ↑_pm cpm 8768 ℂcc 11032 ℝcr 11033 +∞cpnf 11172 ≤ cle 11176 [,)cico 13295 abscabs 15191 𝑂(1)co1 15443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3725 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-fv 6496 df-ov 7362 df-oprab 7363 df-mpo 7364 df-pm 8770 df-o1 15447

This theorem is referenced by: o1res 15517 o1of2 15570 o1rlimmul 15576 o1mptrcl 15580 o1fsum 15771 o1cxp 26959 dchrisum0 27504

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