o1f - Metamath Proof Explorer

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Theorem o1f 15489

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 15486	. . 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 11117	. . . 4 ⊢ ℂ ∈ V
4		reex 11127	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8819	. . 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 2119 ∀wral 3054 ∃wrex 3064 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5079 dom cdm 5625 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ↑_pm cpm 8771 ℂcc 11034 ℝcr 11035 +∞cpnf 11174 ≤ cle 11178 [,)cico 13298 abscabs 15194 𝑂(1)co1 15446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-pm 8773 df-o1 15450

This theorem is referenced by: o1res 15520 o1of2 15573 o1rlimmul 15579 o1mptrcl 15583 o1fsum 15774 o1cxp 26963 dchrisum0 27508

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