o1f - Metamath Proof Explorer

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Theorem o1f 15728

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 15725	. . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
2	1	simplbi 502	. 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11311	. . . 4 ⊢ ℂ ∈ V
4		reex 11321	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8933	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 502	. 2 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
7	2, 6	syl 18	1 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 ∩ cin 4092 class class class wbr 5106 dom cdm 5656 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 ↑_pm cpm 8879 ℂcc 11228 ℝcr 11229 +∞cpnf 11370 ≤ cle 11374 [,)cico 13508 abscabs 15433 𝑂(1)co1 15685

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-pm 8881 df-o1 15689

This theorem is used by: o1res 15759 o1of2 15812 o1rlimmul 15818 o1mptrcl 15822 o1fsum 16012 o1cxp 27335 dchrisum0 27880

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